4.3. Harmonic-analysis method: the affine Mackey model🔗

The affine group is the concrete instance that Theorem 2.5 of arXiv:2405.13682 needs, and it is also where the analysis is hardest. This chapter follows the Lean dependency order of the affine layer: the Jacobian and the two quasi-invariant L^2 representations, the explicit frequency action, the group and orbit input of the Mackey machine, the homogeneous-space model and its quasi-invariant character twist, the normalized-section realization of unitary induction, and the inducing fiber of the imprimitivity argument.

Theorem4.3.1

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Lemma 4.2.6

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Theorem 4.3.2

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✓L∃∀N

Concrete affine instance. Writing g(x)=Lx+t, the parameter action is the block lower-triangular map (a,b)\mapsto(L^{-T}a,b+\langle t,L^{-T}a\rangle). Its determinant is $(\det L)^{-1}. Additive Haar measure therefore has reciprocal data/parameter densities $\lVert\det L\rVert and \lVert\det L\rVert^{-1}; their square roots give both balance laws. The generic quasi-invariant construction now yields the two unitary L^2 representations and the synthesis/ridgelet Bochner intertwining identities for the affine feature.

Lean code for Theorem4.3.1●11 declarations

defdefined in LeanRidgelet/HA/Affine.lean

complete

def LeanRidgelet.affineParameterLinearEquiv.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  (g : E ≃ᵃ[ℝ] E) : (E × ℝ) ≃ₗ[ℝ] E × ℝ

def LeanRidgelet.affineParameterLinearEquiv.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  (g : E ≃ᵃ[ℝ] E) : (E × ℝ) ≃ₗ[ℝ] E × ℝ

Implementation after :=

:=
  g.linear.symm.adjoint.skewProd (LinearEquiv.refl ℝ ℝ) (affineParameterShear g)

The contragredient affine action on ridge parameters `(a, b)`. If `g x = L x + t`, this sends
`(a, b)` to `(L⁻ᵀ a, b + ⟪t, L⁻ᵀ a⟫)`.

theoremdefined in LeanRidgelet/HA/Affine.lean

complete

theorem LeanRidgelet.affineRidgeArgument_invariant.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  (g : E ≃ᵃ[ℝ] E) (x : E) (p : E × ℝ) :
  LeanRidgelet.affineRidgeArgument (g x)
      ((LeanRidgelet.affineParameterLinearEquiv g) p) =
    LeanRidgelet.affineRidgeArgument x p

theorem LeanRidgelet.affineRidgeArgument_invariant.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (g : E ≃ᵃ[ℝ] E)
  (x : E) (p : E × ℝ) :
  LeanRidgelet.affineRidgeArgument (g x)
      ((LeanRidgelet.affineParameterLinearEquiv
          g)
        p) =
    LeanRidgelet.affineRidgeArgument x p

The ridge argument is invariant under the joint affine data/parameter action.

theoremdefined in LeanRidgelet/HA/Affine.lean

complete

theorem LeanRidgelet.det_affineParameterLinearEquiv.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  (g : E ≃ᵃ[ℝ] E) :
  LinearMap.det ↑(LeanRidgelet.affineParameterLinearEquiv g) =
    (LinearMap.det ↑g.linear)⁻¹

theorem LeanRidgelet.det_affineParameterLinearEquiv.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  (g : E ≃ᵃ[ℝ] E) :
  LinearMap.det
      ↑(LeanRidgelet.affineParameterLinearEquiv
          g) =
    (LinearMap.det ↑g.linear)⁻¹

The parameter action has determinant `(det L)⁻¹`.

defdefined in LeanRidgelet/HA/Affine.lean

complete

def LeanRidgelet.affineDataJacobian.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] (g : E ≃ᵃ[ℝ] E) :
  E → NNReal

def LeanRidgelet.affineDataJacobian.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  (g : E ≃ᵃ[ℝ] E) : E → NNReal

Implementation after :=

:=
  fun _ ↦ ‖LinearMap.det (g.linear : E →ₗ[ℝ] E)‖₊

The constant Radon--Nikodym density for the inverse affine data action.

defdefined in LeanRidgelet/HA/Affine.lean

complete

def LeanRidgelet.affineParameterJacobian.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] (g : E ≃ᵃ[ℝ] E) :
  E × ℝ → NNReal

def LeanRidgelet.affineParameterJacobian.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  (g : E ≃ᵃ[ℝ] E) : E × ℝ → NNReal

Implementation after :=

:=
  fun _ ↦ ‖LinearMap.det (g.linear : E →ₗ[ℝ] E)‖₊⁻¹

The constant Radon--Nikodym density for the inverse contragredient parameter action.

theoremdefined in LeanRidgelet/HA/Affine.lean

complete

theorem LeanRidgelet.affine_synthesis_radonNikodym_balance.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] (g : E ≃ᵃ[ℝ] E) (x : E)
  (p : E × ℝ) :
  ↑↑(NNReal.sqrt (LeanRidgelet.affineParameterJacobian g p)) =
    LeanRidgelet.radonNikodymWeight LeanRidgelet.affineDataJacobian g x

theorem LeanRidgelet.affine_synthesis_radonNikodym_balance.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E] (g : E ≃ᵃ[ℝ] E)
  (x : E) (p : E × ℝ) :
  ↑↑(NNReal.sqrt
          (LeanRidgelet.affineParameterJacobian
            g p)) =
    LeanRidgelet.radonNikodymWeight
      LeanRidgelet.affineDataJacobian g x

The synthesis balance in the form expected by the generic quasi-invariant theorem.

theoremdefined in LeanRidgelet/HA/Affine.lean

complete

theorem LeanRidgelet.affine_ridgelet_radonNikodym_balance.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  (g : E ≃ᵃ[ℝ] E) (x : E) (p : E × ℝ) :
  ↑↑(NNReal.sqrt (LeanRidgelet.affineDataJacobian g x)) =
    LeanRidgelet.radonNikodymWeight LeanRidgelet.affineParameterJacobian
      g p

theorem LeanRidgelet.affine_ridgelet_radonNikodym_balance.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (g : E ≃ᵃ[ℝ] E)
  (x : E) (p : E × ℝ) :
  ↑↑(NNReal.sqrt
          (LeanRidgelet.affineDataJacobian
            g x)) =
    LeanRidgelet.radonNikodymWeight
      LeanRidgelet.affineParameterJacobian
      g p

The ridgelet balance in the form expected by the generic quasi-invariant theorem.

defdefined in LeanRidgelet/HA/Affine.lean

complete

def LeanRidgelet.affineDataLpUnitaryRepresentation.{u_1, u_2} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E] {Y : Type u_2}
  [NormedAddCommGroup Y] [InnerProductSpace ℂ Y] [CompleteSpace Y]
  (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure]
  [MeasureTheory.SigmaFinite μ] :
  UnitaryRepresentation (E ≃ᵃ[ℝ] E) ↥(MeasureTheory.Lp Y 2 μ)

def LeanRidgelet.affineDataLpUnitaryRepresentation.{u_1,
    u_2}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  {Y : Type u_2} [NormedAddCommGroup Y]
  [InnerProductSpace ℂ Y]
  [CompleteSpace Y]
  (μ : MeasureTheory.Measure E)
  [μ.IsAddHaarMeasure]
  [MeasureTheory.SigmaFinite μ] :
  UnitaryRepresentation (E ≃ᵃ[ℝ] E)
    ↥(MeasureTheory.Lp Y 2 μ)

Implementation after :=

:=
  quasiInvariantLpUnitaryRepresentation (E := Y) affineDataJacobian
    affineData_measurable (affineData_group_map_eq_withDensity μ)
    affineDataJacobian_measurable affineDataJacobian_ne_zero
    affineDataJacobian_one affineDataJacobian_cocycle

The Radon--Nikodym-corrected affine data representation on Bochner `L²`.

defdefined in LeanRidgelet/HA/Affine.lean

complete

def LeanRidgelet.affineParameterLpUnitaryRepresentation.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E] (ν : MeasureTheory.Measure (E × ℝ))
  [ν.IsAddHaarMeasure] [MeasureTheory.SigmaFinite ν] :
  UnitaryRepresentation (E ≃ᵃ[ℝ] E) ↥(MeasureTheory.Lp ℂ 2 ν)

def LeanRidgelet.affineParameterLpUnitaryRepresentation.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  (ν : MeasureTheory.Measure (E × ℝ))
  [ν.IsAddHaarMeasure]
  [MeasureTheory.SigmaFinite ν] :
  UnitaryRepresentation (E ≃ᵃ[ℝ] E)
    ↥(MeasureTheory.Lp ℂ 2 ν)

Implementation after :=

:=
  quasiInvariantLpUnitaryRepresentation affineParameterJacobian
    affineParameter_measurable (affineParameter_group_map_eq_withDensity ν)
    affineParameterJacobian_measurable affineParameterJacobian_ne_zero
    affineParameterJacobian_one affineParameterJacobian_cocycle

The Radon--Nikodym-corrected contragredient parameter representation on scalar `L²`.

theoremdefined in LeanRidgelet/HA/Affine.lean

complete

theorem LeanRidgelet.affineBochnerSynthesis_intertwines.{u_1, u_2}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
  {Y : Type u_2} [NormedAddCommGroup Y] [InnerProductSpace ℂ Y]
  [CompleteSpace Y] (ν : MeasureTheory.Measure (E × ℝ))
  [ν.IsAddHaarMeasure] (σ : ℝ → Y) (g : E ≃ᵃ[ℝ] E) (γ : E × ℝ → ℂ)
  (x : E) :
  LeanRidgelet.bochnerSynthesis ν (LeanRidgelet.affineFeature σ)
      (LeanRidgelet.quasiRegularAction
        (LeanRidgelet.radonNikodymWeight
          LeanRidgelet.affineParameterJacobian)
        g γ)
      x =
    LeanRidgelet.quasiUnitaryPullbackAction
      LeanRidgelet.affineDataJacobian 1 g
      (LeanRidgelet.bochnerSynthesis ν (LeanRidgelet.affineFeature σ) γ)
      x

theorem LeanRidgelet.affineBochnerSynthesis_intertwines.{u_1,
    u_2}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  {Y : Type u_2} [NormedAddCommGroup Y]
  [InnerProductSpace ℂ Y]
  [CompleteSpace Y]
  (ν : MeasureTheory.Measure (E × ℝ))
  [ν.IsAddHaarMeasure] (σ : ℝ → Y)
  (g : E ≃ᵃ[ℝ] E) (γ : E × ℝ → ℂ)
  (x : E) :
  LeanRidgelet.bochnerSynthesis ν
      (LeanRidgelet.affineFeature σ)
      (LeanRidgelet.quasiRegularAction
        (LeanRidgelet.radonNikodymWeight
          LeanRidgelet.affineParameterJacobian)
        g γ)
      x =
    LeanRidgelet.quasiUnitaryPullbackAction
      LeanRidgelet.affineDataJacobian 1 g
      (LeanRidgelet.bochnerSynthesis ν
        (LeanRidgelet.affineFeature σ) γ)
      x

The quasi-invariant Bochner synthesis identity instantiated for the affine fully-connected
feature.

theoremdefined in LeanRidgelet/HA/Affine.lean

complete

theorem LeanRidgelet.affineBochnerRidgelet_intertwines.{u_1, u_2} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E] {Y : Type u_2}
  [NormedAddCommGroup Y] [InnerProductSpace ℂ Y] [CompleteSpace Y]
  (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] (ψ : ℝ → Y)
  (g : E ≃ᵃ[ℝ] E) (f : E → Y) (p : E × ℝ) :
  LeanRidgelet.bochnerRidgelet μ (LeanRidgelet.affineFeature ψ)
      (LeanRidgelet.quasiUnitaryPullbackAction
        LeanRidgelet.affineDataJacobian 1 g f)
      p =
    LeanRidgelet.quasiRegularAction
      (LeanRidgelet.radonNikodymWeight
        LeanRidgelet.affineParameterJacobian)
      g
      (LeanRidgelet.bochnerRidgelet μ (LeanRidgelet.affineFeature ψ) f)
      p

theorem LeanRidgelet.affineBochnerRidgelet_intertwines.{u_1,
    u_2}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  {Y : Type u_2} [NormedAddCommGroup Y]
  [InnerProductSpace ℂ Y]
  [CompleteSpace Y]
  (μ : MeasureTheory.Measure E)
  [μ.IsAddHaarMeasure] (ψ : ℝ → Y)
  (g : E ≃ᵃ[ℝ] E) (f : E → Y)
  (p : E × ℝ) :
  LeanRidgelet.bochnerRidgelet μ
      (LeanRidgelet.affineFeature ψ)
      (LeanRidgelet.quasiUnitaryPullbackAction
        LeanRidgelet.affineDataJacobian 1
        g f)
      p =
    LeanRidgelet.quasiRegularAction
      (LeanRidgelet.radonNikodymWeight
        LeanRidgelet.affineParameterJacobian)
      g
      (LeanRidgelet.bochnerRidgelet μ
        (LeanRidgelet.affineFeature ψ) f)
      p

The quasi-invariant Bochner ridgelet identity instantiated for the affine fully-connected
feature.

Theorem4.3.2

Statement uses 4

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Theorem 4.3.1

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used by 1✓L∃∀N

The explicit frequency action on all of L². The contragredient action has reciprocal determinant density, so its corrected pullback is f(\xi)\mapsto |\det L|^{1/2}f(L^T\xi). Multiplication by the norm-one translation character is a second unitary operator. Their composite therefore exists directly on Lp classes and has the expected almost-everywhere representative. The Fourier covariance calculation first proves equality with the Plancherel conjugate on Schwartz functions; continuity and Schwartz density extend it to every L² class.

Lean code for Theorem4.3.2●14 declarations

defdefined in LeanRidgelet/HA/AffineFrequency.lean

complete

def LeanRidgelet.affineDualAction.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  (g : E ≃ᵃ[ℝ] E) (xi : E) : E

def LeanRidgelet.affineDualAction.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (g : E ≃ᵃ[ℝ] E)
  (xi : E) : E

Implementation after :=

:= g.linear.symm.adjoint xi

The affine group acts on frequency space through the contragredient of its linear part.

defdefined in LeanRidgelet/HA/AffineFrequency.lean

complete

def LeanRidgelet.affineDualJacobian.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] (g : E ≃ᵃ[ℝ] E) :
  E → NNReal

def LeanRidgelet.affineDualJacobian.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  (g : E ≃ᵃ[ℝ] E) : E → NNReal

Implementation after :=

:=
  fun _ ↦ ‖LinearMap.det (g.linear : E →ₗ[ℝ] E)‖₊⁻¹

The constant density of the inverse dual action.

theoremdefined in LeanRidgelet/HA/AffineFrequency.lean

complete

theorem LeanRidgelet.affineDualAction_map_eq_withDensity.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E] (g : E ≃ᵃ[ℝ] E) :
  MeasureTheory.Measure.map (LeanRidgelet.affineDualAction g⁻¹)
      MeasureTheory.volume =
    MeasureTheory.volume.withDensity fun xi ↦
      ↑(LeanRidgelet.affineDualJacobian g xi)

theorem LeanRidgelet.affineDualAction_map_eq_withDensity.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  (g : E ≃ᵃ[ℝ] E) :
  MeasureTheory.Measure.map
      (LeanRidgelet.affineDualAction g⁻¹)
      MeasureTheory.volume =
    MeasureTheory.volume.withDensity
      fun xi ↦
      ↑(LeanRidgelet.affineDualJacobian g
          xi)

Pushforward of Lebesgue measure by the inverse dual action.

defdefined in LeanRidgelet/HA/AffineFrequency.lean

complete

def LeanRidgelet.affineDualPullbackLpUnitaryRepresentation.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] :
  UnitaryRepresentation (E ≃ᵃ[ℝ] E)
    ↥(MeasureTheory.Lp ℂ 2 MeasureTheory.volume)

def LeanRidgelet.affineDualPullbackLpUnitaryRepresentation.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E] :
  UnitaryRepresentation (E ≃ᵃ[ℝ] E)
    ↥(MeasureTheory.Lp ℂ 2
        MeasureTheory.volume)

Implementation after :=

:= by
  letI : MulAction (E ≃ᵃ[ℝ] E) E := affineDualMulAction
  exact quasiInvariantLpUnitaryRepresentation affineDualJacobian
    (fun g ↦ affineDualAction_measurable g)
    (fun g ↦ affineDualAction_map_eq_withDensity g)
    affineDualJacobian_measurable affineDualJacobian_ne_zero
    affineDualJacobian_one (fun g h xi ↦ affineDualJacobian_cocycle g h xi)

The determinant-corrected pullback representation for the contragredient action on frequency
space.

theoremdefined in LeanRidgelet/HA/AffineFrequency.lean

complete

theorem LeanRidgelet.affineDualPullbackLpUnitaryRepresentation_apply_ae.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
  (g : E ≃ᵃ[ℝ] E) (f : ↥(MeasureTheory.Lp ℂ 2 MeasureTheory.volume)) :
  ↑↑(↑(LeanRidgelet.affineDualPullbackLpUnitaryRepresentation g)
          f) =ᵐ[MeasureTheory.volume]
    fun xi ↦
    ↑↑(NNReal.sqrt ‖LinearMap.det ↑g.linear‖₊) •
      ↑↑f (g.linear.adjoint xi)

theorem LeanRidgelet.affineDualPullbackLpUnitaryRepresentation_apply_ae.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  (g : E ≃ᵃ[ℝ] E)
  (f :
    ↥(MeasureTheory.Lp ℂ 2
        MeasureTheory.volume)) :
  ↑↑(↑(LeanRidgelet.affineDualPullbackLpUnitaryRepresentation
              g)
          f) =ᵐ[MeasureTheory.volume]
    fun xi ↦
    ↑↑(NNReal.sqrt
            ‖LinearMap.det ↑g.linear‖₊) •
      ↑↑f (g.linear.adjoint xi)

The dual pullback representation acts by `|det L|¹ᐟ² f(Lᵀξ)` on every `L²` class.

defdefined in LeanRidgelet/HA/AffineFrequency.lean

complete

def LeanRidgelet.affineFrequencyPhase.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] (g : E ≃ᵃ[ℝ] E)
  (xi : E) : ℂ

def LeanRidgelet.affineFrequencyPhase.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E] (g : E ≃ᵃ[ℝ] E)
  (xi : E) : ℂ

Implementation after :=

:=
  ((Real.fourierChar (-⟪g 0, xi⟫_ℝ) : Circle) : ℂ)

The translation character appearing in the Fourier-side affine action.

theoremdefined in LeanRidgelet/HA/AffineFrequency.lean

complete

theorem LeanRidgelet.affineFrequencyPhase_translation.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] (b xi : E) :
  LeanRidgelet.affineFrequencyPhase (LeanRidgelet.affineTranslation b)
      xi =
    ↑((LeanRidgelet.affineTranslationCharacter xi) b)

theorem LeanRidgelet.affineFrequencyPhase_translation.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E] (b xi : E) :
  LeanRidgelet.affineFrequencyPhase
      (LeanRidgelet.affineTranslation b)
      xi =
    ↑((LeanRidgelet.affineTranslationCharacter
          xi)
        b)

On the translation subgroup, the Fourier-side phase is precisely the Mackey translation
character at the current frequency.

theoremdefined in LeanRidgelet/HA/AffineFrequency.lean

complete

theorem LeanRidgelet.affineFrequencyPhase_cocycle.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  (g h : E ≃ᵃ[ℝ] E) (xi : E) :
  LeanRidgelet.affineFrequencyPhase (g * h) xi =
    LeanRidgelet.affineFrequencyPhase g xi *
      LeanRidgelet.affineFrequencyPhase h
        (LeanRidgelet.affineDualAction g⁻¹ xi)

theorem LeanRidgelet.affineFrequencyPhase_cocycle.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  (g h : E ≃ᵃ[ℝ] E) (xi : E) :
  LeanRidgelet.affineFrequencyPhase
      (g * h) xi =
    LeanRidgelet.affineFrequencyPhase g
        xi *
      LeanRidgelet.affineFrequencyPhase h
        (LeanRidgelet.affineDualAction g⁻¹
          xi)

The affine frequency phase is a multiplier cocycle for inverse dual pullback.  This is the
character twist that distinguishes the Mackey representation from the untwisted quasi-regular
action on the homogeneous orbit.

defdefined in LeanRidgelet/HA/AffineFrequency.lean

complete

def LeanRidgelet.affineFrequencyPhaseMultiplier.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E] (g : E ≃ᵃ[ℝ] E) :
  ↥(MeasureTheory.Lp ℂ 2 MeasureTheory.volume) ≃ₗᵢ[ℂ]
    ↥(MeasureTheory.Lp ℂ 2 MeasureTheory.volume)

def LeanRidgelet.affineFrequencyPhaseMultiplier.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  (g : E ≃ᵃ[ℝ] E) :
  ↥(MeasureTheory.Lp ℂ 2
        MeasureTheory.volume) ≃ₗᵢ[ℂ]
    ↥(MeasureTheory.Lp ℂ 2
        MeasureTheory.volume)

Implementation after :=

:=
  unimodularMultiplierLinearIsometryEquiv (affineFrequencyPhase g)
    (affineFrequencyPhase_aestronglyMeasurable g) (affineFrequencyPhase_norm_one g)

Multiplication by the affine frequency phase, bundled as a unitary `Lp` operator.

defdefined in LeanRidgelet/HA/AffineFrequency.lean

complete

def LeanRidgelet.affineFrequencyLinearIsometryEquiv.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E] (g : E ≃ᵃ[ℝ] E) :
  ↥(MeasureTheory.Lp ℂ 2 MeasureTheory.volume) ≃ₗᵢ[ℂ]
    ↥(MeasureTheory.Lp ℂ 2 MeasureTheory.volume)

def LeanRidgelet.affineFrequencyLinearIsometryEquiv.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  (g : E ≃ᵃ[ℝ] E) :
  ↥(MeasureTheory.Lp ℂ 2
        MeasureTheory.volume) ≃ₗᵢ[ℂ]
    ↥(MeasureTheory.Lp ℂ 2
        MeasureTheory.volume)

Implementation after :=

:=
  (Unitary.linearIsometryEquiv (affineDualPullbackLpUnitaryRepresentation (E := E) g)).trans
    (affineFrequencyPhaseMultiplier g)

The explicit Fourier-side affine operator: dual pullback followed by multiplication by the
translation character.

theoremdefined in LeanRidgelet/HA/AffineFrequency.lean

complete

theorem LeanRidgelet.affineFrequencyLinearIsometryEquiv_apply_ae.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
  (g : E ≃ᵃ[ℝ] E) (f : ↥(MeasureTheory.Lp ℂ 2 MeasureTheory.volume)) :
  ↑↑((LeanRidgelet.affineFrequencyLinearIsometryEquiv g)
          f) =ᵐ[MeasureTheory.volume]
    fun xi ↦
    LeanRidgelet.affineFrequencyPhase g xi *
      (↑↑(NNReal.sqrt ‖LinearMap.det ↑g.linear‖₊) *
        ↑↑f (g.linear.adjoint xi))

theorem LeanRidgelet.affineFrequencyLinearIsometryEquiv_apply_ae.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  (g : E ≃ᵃ[ℝ] E)
  (f :
    ↥(MeasureTheory.Lp ℂ 2
        MeasureTheory.volume)) :
  ↑↑((LeanRidgelet.affineFrequencyLinearIsometryEquiv
            g)
          f) =ᵐ[MeasureTheory.volume]
    fun xi ↦
    LeanRidgelet.affineFrequencyPhase g
        xi *
      (↑↑(NNReal.sqrt
              ‖LinearMap.det
                  ↑g.linear‖₊) *
        ↑↑f (g.linear.adjoint xi))

The explicit Fourier-side affine operator acts by character multiplication and adjoint
pullback on every `L²` class.

theoremdefined in LeanRidgelet/HA/AffineFrequency.lean

complete

theorem LeanRidgelet.affineFourierLpUnitaryRepresentation_schwartz_input_ae.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
  (g : E ≃ᵃ[ℝ] E) (f : SchwartzMap E ℂ) :
  ↑↑((Unitary.linearIsometryEquiv
            (LeanRidgelet.affineFourierLpUnitaryRepresentation g))
          (f.toLp 2 MeasureTheory.volume)) =ᵐ[MeasureTheory.volume]
    fun xi ↦
    Real.fourierChar (-inner ℝ (g 0) xi) •
      ↑↑(NNReal.sqrt ‖LinearMap.det ↑g.linear‖₊) •
        f (g.linear.adjoint xi)

theorem LeanRidgelet.affineFourierLpUnitaryRepresentation_schwartz_input_ae.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  (g : E ≃ᵃ[ℝ] E) (f : SchwartzMap E ℂ) :
  ↑↑((Unitary.linearIsometryEquiv
            (LeanRidgelet.affineFourierLpUnitaryRepresentation
              g))
          (f.toLp 2
            MeasureTheory.volume)) =ᵐ[MeasureTheory.volume]
    fun xi ↦
    Real.fourierChar (-inner ℝ (g 0) xi) •
      ↑↑(NNReal.sqrt
              ‖LinearMap.det
                  ↑g.linear‖₊) •
        f (g.linear.adjoint xi)

The Plancherel-conjugated representation has the explicit affine frequency formula on the
ordinary Schwartz core (rather than only on Fourier transforms of Schwartz vectors).

theoremdefined in LeanRidgelet/HA/AffineFrequency.lean

complete

theorem LeanRidgelet.affineFrequencyLinearIsometryEquiv_eq_fourierRepresentation.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
  (g : E ≃ᵃ[ℝ] E) :
  LeanRidgelet.affineFrequencyLinearIsometryEquiv g =
    Unitary.linearIsometryEquiv
      (LeanRidgelet.affineFourierLpUnitaryRepresentation g)

theorem LeanRidgelet.affineFrequencyLinearIsometryEquiv_eq_fourierRepresentation.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  (g : E ≃ᵃ[ℝ] E) :
  LeanRidgelet.affineFrequencyLinearIsometryEquiv
      g =
    Unitary.linearIsometryEquiv
      (LeanRidgelet.affineFourierLpUnitaryRepresentation
        g)

The explicit frequency operator is exactly the Plancherel-conjugated affine action on all of
`L²`.  The proof uses the explicit formulas on the dense Schwartz core and continuity of both
unitary operators.

theoremdefined in LeanRidgelet/HA/AffineFrequency.lean

complete

theorem LeanRidgelet.affineFourierLpUnitaryRepresentation_apply_ae.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
  (g : E ≃ᵃ[ℝ] E) (f : ↥(MeasureTheory.Lp ℂ 2 MeasureTheory.volume)) :
  ↑↑((Unitary.linearIsometryEquiv
            (LeanRidgelet.affineFourierLpUnitaryRepresentation g))
          f) =ᵐ[MeasureTheory.volume]
    fun xi ↦
    LeanRidgelet.affineFrequencyPhase g xi *
      (↑↑(NNReal.sqrt ‖LinearMap.det ↑g.linear‖₊) *
        ↑↑f (g.linear.adjoint xi))

theorem LeanRidgelet.affineFourierLpUnitaryRepresentation_apply_ae.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  (g : E ≃ᵃ[ℝ] E)
  (f :
    ↥(MeasureTheory.Lp ℂ 2
        MeasureTheory.volume)) :
  ↑↑((Unitary.linearIsometryEquiv
            (LeanRidgelet.affineFourierLpUnitaryRepresentation
              g))
          f) =ᵐ[MeasureTheory.volume]
    fun xi ↦
    LeanRidgelet.affineFrequencyPhase g
        xi *
      (↑↑(NNReal.sqrt
              ‖LinearMap.det
                  ↑g.linear‖₊) *
        ↑↑f (g.linear.adjoint xi))

Consequently, the Plancherel-conjugated affine representation has the character-times-dual-
pullback formula for every `L²` class.

Theorem4.3.3

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✓L∃∀N

Group, Fourier, and orbit input for Mackey induction. The affine group is realized as a locally compact topological semidirect product, and the determinant-corrected physical representation is strongly continuous. Plancherel conjugation gives an equivalent frequency representation with bounded intertwiners in both directions; the explicit dual-pullback and character model agrees with it on all of L². The contragredient action has one conull nonzero orbit. At a chosen frequency, the translation group semidirect the little group is a closed locally compact inducing subgroup carrying a continuous irreducible character. The orbit map descends to an equivariant homeomorphism from its left-coset space to that orbit. Pullback to the orbit subtype is a surjective L² isometry, so physical, frequency, and intrinsic-orbit irreducibility are equivalent. The measure, section, and commutant layers are separated below. The full linear group is essential: positive dilations alone leave two frequency half-lines.

Lean code for Theorem4.3.3●18 declarations

defdefined in LeanRidgelet/HA/AffineIrreducibility.lean

complete

def LeanRidgelet.affineTopologicalLpUnitaryRepresentation.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
  (mu : MeasureTheory.Measure E) [mu.IsAddHaarMeasure]
  [MeasureTheory.SigmaFinite mu] :
  UnitaryRepresentation (AffineEquiv.TopologicalSemidirectProduct E)
    ↥(MeasureTheory.Lp ℂ 2 mu)

def LeanRidgelet.affineTopologicalLpUnitaryRepresentation.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  (mu : MeasureTheory.Measure E)
  [mu.IsAddHaarMeasure]
  [MeasureTheory.SigmaFinite mu] :
  UnitaryRepresentation
    (AffineEquiv.TopologicalSemidirectProduct
      E)
    ↥(MeasureTheory.Lp ℂ 2 mu)

Implementation after :=

:=
  (affineDataLpUnitaryRepresentation (Y := ℂ) mu).restrict
    (AffineEquiv.topologicalSemidirectProductEquiv E).toMonoidHom

The affine data representation, reindexed by the locally compact semidirect-product model of
the affine group.  This is the group model to which locally compact Mackey theory applies.

theoremdefined in LeanRidgelet/HA/AffineIrreducibility.lean

complete

theorem LeanRidgelet.affineTopologicalLpUnitaryRepresentation_isStronglyContinuous.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
  (mu : MeasureTheory.Measure E) [mu.IsAddHaarMeasure]
  [MeasureTheory.SigmaFinite mu] [mu.InnerRegularCompactLTTop]
  [MeasureTheory.IsLocallyFiniteMeasure mu] :
  (LeanRidgelet.affineTopologicalLpUnitaryRepresentation
      mu).IsStronglyContinuous

theorem LeanRidgelet.affineTopologicalLpUnitaryRepresentation_isStronglyContinuous.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  (mu : MeasureTheory.Measure E)
  [mu.IsAddHaarMeasure]
  [MeasureTheory.SigmaFinite mu]
  [mu.InnerRegularCompactLTTop]
  [MeasureTheory.IsLocallyFiniteMeasure
      mu] :
  (LeanRidgelet.affineTopologicalLpUnitaryRepresentation
      mu).IsStronglyContinuous

The determinant-corrected affine `L²` representation is strongly continuous on the locally
compact semidirect-product model of the affine group.

defdefined in LeanRidgelet/HA/AffineFourier.lean

complete

def LeanRidgelet.affineTopologicalFourierLpUnitaryRepresentation.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] :
  UnitaryRepresentation (AffineEquiv.TopologicalSemidirectProduct E)
    ↥(MeasureTheory.Lp ℂ 2 MeasureTheory.volume)

def LeanRidgelet.affineTopologicalFourierLpUnitaryRepresentation.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E] :
  UnitaryRepresentation
    (AffineEquiv.TopologicalSemidirectProduct
      E)
    ↥(MeasureTheory.Lp ℂ 2
        MeasureTheory.volume)

Implementation after :=

:=
  (affineTopologicalLpUnitaryRepresentation (volume : Measure E)).conjugate
    (Lp.fourierTransformₗᵢ E ℂ)

The Fourier-side affine representation indexed by the locally compact topological
semidirect-product model.

defdefined in LeanRidgelet/HA/AffineFourier.lean

complete

def LeanRidgelet.affinePlancherelIntertwiningMap.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E] :
  ContIntertwiningMap
    (LeanRidgelet.affineDataLpUnitaryRepresentation
        MeasureTheory.volume).toContRepresentation
    LeanRidgelet.affineFourierLpUnitaryRepresentation.toContRepresentation

def LeanRidgelet.affinePlancherelIntertwiningMap.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E] :
  ContIntertwiningMap
    (LeanRidgelet.affineDataLpUnitaryRepresentation
        MeasureTheory.volume).toContRepresentation
    LeanRidgelet.affineFourierLpUnitaryRepresentation.toContRepresentation

Implementation after :=

:= ℂ) (volume : Measure E)).toContRepresentation →ⁱL
      (affineFourierLpUnitaryRepresentation (E := E)).toContRepresentation :=
  UnitaryRepresentation.conjugateIntertwiningMap
    (affineDataLpUnitaryRepresentation (Y := ℂ) (volume : Measure E))
    (Lp.fourierTransformₗᵢ E ℂ)

The Plancherel transform, bundled as a bounded intertwiner from the physical-space affine
representation to its frequency-space conjugate.

defdefined in LeanRidgelet/HA/AffineFourier.lean

complete

def LeanRidgelet.affinePlancherelInverseIntertwiningMap.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E] :
  ContIntertwiningMap
    LeanRidgelet.affineFourierLpUnitaryRepresentation.toContRepresentation
    (LeanRidgelet.affineDataLpUnitaryRepresentation
        MeasureTheory.volume).toContRepresentation

def LeanRidgelet.affinePlancherelInverseIntertwiningMap.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E] :
  ContIntertwiningMap
    LeanRidgelet.affineFourierLpUnitaryRepresentation.toContRepresentation
    (LeanRidgelet.affineDataLpUnitaryRepresentation
        MeasureTheory.volume).toContRepresentation

Implementation after :=

:= E)).toContRepresentation →ⁱL
      (affineDataLpUnitaryRepresentation (Y := ℂ)
        (volume : Measure E)).toContRepresentation :=
  UnitaryRepresentation.conjugateInverseIntertwiningMap
    (affineDataLpUnitaryRepresentation (Y := ℂ) (volume : Measure E))
    (Lp.fourierTransformₗᵢ E ℂ)

The inverse Plancherel transform, bundled as the inverse bounded intertwiner.

defdefined in LeanRidgelet/HA/AffineIrreducibility.lean

complete

def LeanRidgelet.affineDualOrbit.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] : Set E

def LeanRidgelet.affineDualOrbit.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] :
  Set E

Implementation after :=

:= {xi | xi ≠ 0}

The unique nonzero orbit for the dual action of the full affine linear subgroup.

theoremdefined in LeanRidgelet/HA/AffineIrreducibility.lean

complete

theorem LeanRidgelet.affineDualOrbit_transitive.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  {xi eta : E} (hxi : xi ∈ LeanRidgelet.affineDualOrbit)
  (heta : eta ∈ LeanRidgelet.affineDualOrbit) :
  ∃ L, L.symm.adjoint xi = eta

theorem LeanRidgelet.affineDualOrbit_transitive.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {xi eta : E}
  (hxi :
    xi ∈ LeanRidgelet.affineDualOrbit)
  (heta :
    eta ∈ LeanRidgelet.affineDualOrbit) :
  ∃ L, L.symm.adjoint xi = eta

The contragredient `GL(E)` action is transitive on the affine dual orbit.

defdefined in LeanRidgelet/HA/AffineIrreducibility.lean

complete

def LeanRidgelet.affineTopologicalMackeySubgroup.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  (xi : E) : ClosedSubgroup (AffineEquiv.TopologicalSemidirectProduct E)

def LeanRidgelet.affineTopologicalMackeySubgroup.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (xi : E) :
  ClosedSubgroup
    (AffineEquiv.TopologicalSemidirectProduct
      E)

Implementation after :=

:= (affineTopologicalDualLittleGroup xi).comap SemidirectProduct.rightHom
  isClosed' := (affineTopologicalDualLittleGroup xi).isClosed'.preimage
    (SemidirectProduct.continuous_right
      (φ := AffineEquiv.continuousLinearMultiplicativeActionHom E))

The Mackey inducing subgroup `E ⋊ G_xi` inside the topological affine group. It contains all
translations and restricts the linear factor to the stabilizer of the frequency `xi`.

defdefined in LeanRidgelet/HA/AffineIrreducibility.lean

complete

def LeanRidgelet.affineTopologicalMackeyCharacter.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  (xi : E) :
  ↥(LeanRidgelet.affineTopologicalMackeySubgroup xi) →* Circle

def LeanRidgelet.affineTopologicalMackeyCharacter.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (xi : E) :
  ↥(LeanRidgelet.affineTopologicalMackeySubgroup
        xi) →*
    Circle

Implementation after :=

:= affineTranslationCharacter xi p.1.left.toAdd
  map_one' := by simp
  map_mul' p q := by
    change affineTranslationCharacter xi
        (p.1.left.toAdd + (p.1.right : E →L[ℝ] E) q.1.left.toAdd) =
      affineTranslationCharacter xi p.1.left.toAdd *
        affineTranslationCharacter xi q.1.left.toAdd
    rw [(affineTranslationCharacter xi).map_add_eq_mul]
    congr 1
    exact affineTranslationCharacter_topological_linear_apply p.property q.1.left.toAdd

The character of the inducing subgroup: it is the frequency character on translations and
is trivial on the little group.

defdefined in LeanRidgelet/HA/AffineIrreducibility.lean

complete

def LeanRidgelet.affineTopologicalMackeyUnitaryRepresentation.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (xi : E) :
  UnitaryRepresentation
    ↥(LeanRidgelet.affineTopologicalMackeySubgroup xi) ℂ

def LeanRidgelet.affineTopologicalMackeyUnitaryRepresentation.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (xi : E) :
  UnitaryRepresentation
    ↥(LeanRidgelet.affineTopologicalMackeySubgroup
        xi)
    ℂ

Implementation after :=

:=
  UnitaryRepresentation.ofCircleCharacter (affineTopologicalMackeyCharacter xi)

The one-dimensional unitary representation of the Mackey inducing subgroup.

theoremdefined in LeanRidgelet/HA/AffineIrreducibility.lean

complete

theorem LeanRidgelet.affineTopologicalMackeyUnitaryRepresentation_isTopologicallyIrreducible.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (xi : E) :
  (LeanRidgelet.affineTopologicalMackeyUnitaryRepresentation
      xi).IsTopologicallyIrreducible

theorem LeanRidgelet.affineTopologicalMackeyUnitaryRepresentation_isTopologicallyIrreducible.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (xi : E) :
  (LeanRidgelet.affineTopologicalMackeyUnitaryRepresentation
      xi).IsTopologicallyIrreducible

The Mackey inducing representation is irreducible because its Hilbert space is the complex
line.

defdefined in LeanRidgelet/HA/AffineIrreducibility.lean

complete

def LeanRidgelet.affineTopologicalMackeyQuotientHomeomorphDualOrbit.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {xi : E} (hxi : xi ≠ 0) :
  AffineEquiv.TopologicalSemidirectProduct E ⧸
      ↑(LeanRidgelet.affineTopologicalMackeySubgroup xi) ≃ₜ
    ↑LeanRidgelet.affineDualOrbit

def LeanRidgelet.affineTopologicalMackeyQuotientHomeomorphDualOrbit.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {xi : E}
  (hxi : xi ≠ 0) :
  AffineEquiv.TopologicalSemidirectProduct
        E ⧸
      ↑(LeanRidgelet.affineTopologicalMackeySubgroup
          xi) ≃ₜ
    ↑LeanRidgelet.affineDualOrbit

Implementation after :=

:= E) :=
  (affineTopologicalMackeyQuotientEquivDualOrbit hxi).toHomeomorphOfContinuousOpen
    (continuous_affineTopologicalMackeyQuotientEquivDualOrbit hxi)
    (isOpenMap_affineTopologicalMackeyQuotientOrbitMap hxi)

The homogeneous left-coset space is homeomorphic to the nonzero dual orbit.

defdefined in LeanRidgelet/HA/AffineMackeyMeasure.lean

complete

def LeanRidgelet.affineDualOrbitSubtypeMeasure.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E] :
  MeasureTheory.Measure ↑LeanRidgelet.affineDualOrbit

def LeanRidgelet.affineDualOrbitSubtypeMeasure.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E] :
  MeasureTheory.Measure
    ↑LeanRidgelet.affineDualOrbit

Implementation after :=

:= E)) :=
  Measure.comap Subtype.val (volume : Measure E)

The intrinsic measure on the nonzero-frequency subtype, obtained by pulling Lebesgue measure
back along the subtype inclusion.

defdefined in LeanRidgelet/HA/AffineMackeyMeasure.lean

complete

def LeanRidgelet.affineDualOrbitSubtypeLpEquiv.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [Nontrivial E] [MeasurableSpace E] [BorelSpace E] :
  ↥(MeasureTheory.Lp ℂ 2 MeasureTheory.volume) ≃ₗᵢ[ℂ]
    ↥(MeasureTheory.Lp ℂ 2 LeanRidgelet.affineDualOrbitSubtypeMeasure)

def LeanRidgelet.affineDualOrbitSubtypeLpEquiv.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E]
  [MeasurableSpace E] [BorelSpace E] :
  ↥(MeasureTheory.Lp ℂ 2
        MeasureTheory.volume) ≃ₗᵢ[ℂ]
    ↥(MeasureTheory.Lp ℂ 2
        LeanRidgelet.affineDualOrbitSubtypeMeasure)

Implementation after :=

:= E)) :=
  LinearIsometryEquiv.ofSurjective (affineDualOrbitRestrictionLpLinearIsometry (E := E))
    affineDualOrbitRestrictionLpLinearIsometry_surjective

The full frequency `L²` space and the intrinsic `L²` space on the nonzero-frequency subtype
are canonically linearly isometric.

defdefined in LeanRidgelet/HA/AffineMackey.lean

complete

def LeanRidgelet.affineTopologicalOrbitLpUnitaryRepresentation.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E] [MeasurableSpace E]
  [BorelSpace E] :
  UnitaryRepresentation (AffineEquiv.TopologicalSemidirectProduct E)
    ↥(MeasureTheory.Lp ℂ 2 LeanRidgelet.affineDualOrbitSubtypeMeasure)

def LeanRidgelet.affineTopologicalOrbitLpUnitaryRepresentation.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E]
  [MeasurableSpace E] [BorelSpace E] :
  UnitaryRepresentation
    (AffineEquiv.TopologicalSemidirectProduct
      E)
    ↥(MeasureTheory.Lp ℂ 2
        LeanRidgelet.affineDualOrbitSubtypeMeasure)

Implementation after :=

:= E))) :=
  (affineTopologicalFourierLpUnitaryRepresentation (E := E)).conjugate
    (affineDualOrbitSubtypeLpEquiv (E := E))

The explicit Fourier-side affine representation transported to the intrinsic `L²` space on
the nonzero-frequency subtype.

defdefined in LeanRidgelet/HA/AffineMackey.lean

complete

def LeanRidgelet.affineDualOrbitIntertwiningMap.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [Nontrivial E] [MeasurableSpace E] [BorelSpace E] :
  ContIntertwiningMap
    LeanRidgelet.affineTopologicalFourierLpUnitaryRepresentation.toContRepresentation
    LeanRidgelet.affineTopologicalOrbitLpUnitaryRepresentation.toContRepresentation

def LeanRidgelet.affineDualOrbitIntertwiningMap.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E]
  [MeasurableSpace E] [BorelSpace E] :
  ContIntertwiningMap
    LeanRidgelet.affineTopologicalFourierLpUnitaryRepresentation.toContRepresentation
    LeanRidgelet.affineTopologicalOrbitLpUnitaryRepresentation.toContRepresentation

Implementation after :=

:= E)).toContRepresentation →ⁱL
      (affineTopologicalOrbitLpUnitaryRepresentation (E := E)).toContRepresentation :=
  UnitaryRepresentation.conjugateIntertwiningMap
    (affineTopologicalFourierLpUnitaryRepresentation (E := E))
    (affineDualOrbitSubtypeLpEquiv (E := E))

The full frequency-space/orbit-space identification as a bounded intertwiner.

defdefined in LeanRidgelet/HA/AffineMackey.lean

complete

def LeanRidgelet.affineDualOrbitInverseIntertwiningMap.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [Nontrivial E] [MeasurableSpace E] [BorelSpace E] :
  ContIntertwiningMap
    LeanRidgelet.affineTopologicalOrbitLpUnitaryRepresentation.toContRepresentation
    LeanRidgelet.affineTopologicalFourierLpUnitaryRepresentation.toContRepresentation

def LeanRidgelet.affineDualOrbitInverseIntertwiningMap.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E]
  [MeasurableSpace E] [BorelSpace E] :
  ContIntertwiningMap
    LeanRidgelet.affineTopologicalOrbitLpUnitaryRepresentation.toContRepresentation
    LeanRidgelet.affineTopologicalFourierLpUnitaryRepresentation.toContRepresentation

Implementation after :=

:= E)).toContRepresentation →ⁱL
      (affineTopologicalFourierLpUnitaryRepresentation (E := E)).toContRepresentation :=
  UnitaryRepresentation.conjugateInverseIntertwiningMap
    (affineTopologicalFourierLpUnitaryRepresentation (E := E))
    (affineDualOrbitSubtypeLpEquiv (E := E))

The inverse orbit/full-frequency identification as a bounded intertwiner.

theoremdefined in LeanRidgelet/HA/AffineMackey.lean

complete

theorem LeanRidgelet.affineTopologicalOrbitLpUnitaryRepresentation_isTopologicallyIrreducible_iff.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E] [MeasurableSpace E]
  [BorelSpace E] :
  LeanRidgelet.affineTopologicalOrbitLpUnitaryRepresentation.IsTopologicallyIrreducible ↔
    (LeanRidgelet.affineDataLpUnitaryRepresentation
        MeasureTheory.volume).IsTopologicallyIrreducible

theorem LeanRidgelet.affineTopologicalOrbitLpUnitaryRepresentation_isTopologicallyIrreducible_iff.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E]
  [MeasurableSpace E] [BorelSpace E] :
  LeanRidgelet.affineTopologicalOrbitLpUnitaryRepresentation.IsTopologicallyIrreducible ↔
    (LeanRidgelet.affineDataLpUnitaryRepresentation
        MeasureTheory.volume).IsTopologicallyIrreducible

The orbit-measure, Fourier, and physical affine models are topologically irreducible
simultaneously.

The homogeneous-space L² model. The quotient-orbit homeomorphism intertwines left translation on G/H with the affine dual action. The affine semidirect product is Polish and the closed subgroup quotient is second countable and Borel. Transporting the intrinsic orbit measure back through the homeomorphism gives a measure on the actual left-coset space; the homeomorphism and its inverse are measure preserving. Both transported measures are finite on compact sets, so compactly supported continuous sections belong to L². Mathlib's composition API gives an onto linear isometry between orbit L^2 and homogeneous-space L^2, with inverse pullback supplying the surjectivity proof. Conjugation transports the explicit orbit representation to L^2(G/H) and bundles the equivalence and inverse as bounded intertwiners. Strong continuity and equivalence of irreducibility with the physical model follow without a measurable section. The normalized-section realization is constructed in the next node; only its spectral/commutant theorem remains afterward.

Lean code for Theorem4.3.4●21 declarations

theoremdefined in LeanRidgelet/HA/AffineIrreducibility.lean

complete

theorem LeanRidgelet.instIsClosedAffineTopologicalMackeySubgroup.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (xi : E) :
  IsClosed ↑↑(LeanRidgelet.affineTopologicalMackeySubgroup xi)

theorem LeanRidgelet.instIsClosedAffineTopologicalMackeySubgroup.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (xi : E) :
  IsClosed
    ↑↑(LeanRidgelet.affineTopologicalMackeySubgroup
          xi)

The underlying subgroup of the closed Mackey inducing subgroup is closed.  This instance lets
Mathlib equip its homogeneous coset space with the separated quotient topology.

theoremdefined in LeanRidgelet/HA/AffineIrreducibility.lean

complete

theorem LeanRidgelet.instSecondCountableTopologyAffineTopologicalMackeyQuotient.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (xi : E) :
  SecondCountableTopology
    (AffineEquiv.TopologicalSemidirectProduct E ⧸
      ↑(LeanRidgelet.affineTopologicalMackeySubgroup xi))

theorem LeanRidgelet.instSecondCountableTopologyAffineTopologicalMackeyQuotient.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (xi : E) :
  SecondCountableTopology
    (AffineEquiv.TopologicalSemidirectProduct
        E ⧸
      ↑(LeanRidgelet.affineTopologicalMackeySubgroup
          xi))

The homogeneous coset space of the affine group is second countable.  Recording the direct
quotient instance avoids making downstream Borel-space synthesis rediscover second countability of
the general linear factor.

theoremdefined in LeanRidgelet/HA/AffineIrreducibility.lean

complete

theorem LeanRidgelet.affineTopologicalMackeyQuotientHomeomorphDualOrbit_smul.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {xi : E} (hxi : xi ≠ 0)
  (g : AffineEquiv.TopologicalSemidirectProduct E)
  (q :
    AffineEquiv.TopologicalSemidirectProduct E ⧸
      ↑(LeanRidgelet.affineTopologicalMackeySubgroup xi)) :
  (LeanRidgelet.affineTopologicalMackeyQuotientHomeomorphDualOrbit hxi)
      (g • q) =
    g •
      (LeanRidgelet.affineTopologicalMackeyQuotientHomeomorphDualOrbit
          hxi)
        q

theorem LeanRidgelet.affineTopologicalMackeyQuotientHomeomorphDualOrbit_smul.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {xi : E}
  (hxi : xi ≠ 0)
  (g :
    AffineEquiv.TopologicalSemidirectProduct
      E)
  (q :
    AffineEquiv.TopologicalSemidirectProduct
        E ⧸
      ↑(LeanRidgelet.affineTopologicalMackeySubgroup
          xi)) :
  (LeanRidgelet.affineTopologicalMackeyQuotientHomeomorphDualOrbit
        hxi)
      (g • q) =
    g •
      (LeanRidgelet.affineTopologicalMackeyQuotientHomeomorphDualOrbit
          hxi)
        q

The quotient-orbit homeomorphism intertwines left translation on the homogeneous space with
the affine dual action on the nonzero orbit.

abbrevdefined in LeanRidgelet/HA/AffineMackeyMeasure.lean

complete

abbrev LeanRidgelet.AffineTopologicalMackeyQuotient.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  (xi : E) : Type u_1

abbrev LeanRidgelet.AffineTopologicalMackeyQuotient.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (xi : E) :
  Type u_1

Implementation after :=

:=
  AffineEquiv.TopologicalSemidirectProduct E ⧸
    (affineTopologicalMackeySubgroup xi).toSubgroup

The homogeneous left-coset space of the full affine group by the Mackey inducing subgroup at
the frequency `xi`.

defdefined in LeanRidgelet/HA/AffineMackeyMeasure.lean

complete

def LeanRidgelet.affineTopologicalMackeyQuotientMeasure.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E] {xi : E} (hxi : xi ≠ 0) :
  MeasureTheory.Measure
    (LeanRidgelet.AffineTopologicalMackeyQuotient xi)

def LeanRidgelet.affineTopologicalMackeyQuotientMeasure.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  {xi : E} (hxi : xi ≠ 0) :
  MeasureTheory.Measure
    (LeanRidgelet.AffineTopologicalMackeyQuotient
      xi)

Implementation after :=

:=
  (affineDualOrbitSubtypeMeasure (E := E)).map
    (affineTopologicalMackeyQuotientHomeomorphDualOrbit hxi).symm

The measure on the affine homogeneous space obtained by transporting the intrinsic Lebesgue
measure of the nonzero dual orbit through the quotient-orbit homeomorphism.

theoremdefined in LeanRidgelet/HA/AffineMackeyMeasure.lean

complete

theorem LeanRidgelet.instIsFiniteMeasureOnCompactsAffineDualOrbitSubtypeMeasure.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] :
  MeasureTheory.IsFiniteMeasureOnCompacts
    LeanRidgelet.affineDualOrbitSubtypeMeasure

theorem LeanRidgelet.instIsFiniteMeasureOnCompactsAffineDualOrbitSubtypeMeasure.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E] :
  MeasureTheory.IsFiniteMeasureOnCompacts
    LeanRidgelet.affineDualOrbitSubtypeMeasure

The intrinsic orbit measure is finite on compact sets.  This is the measure-theoretic input
needed to send compactly supported continuous orbit sections to `L²`.

theoremdefined in LeanRidgelet/HA/AffineMackeyMeasure.lean

complete

theorem LeanRidgelet.instIsFiniteMeasureOnCompactsAffineTopologicalMackeyQuotientMeasure.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] {xi : E}
  (hxi : xi ≠ 0) :
  MeasureTheory.IsFiniteMeasureOnCompacts
    (LeanRidgelet.affineTopologicalMackeyQuotientMeasure hxi)

theorem LeanRidgelet.instIsFiniteMeasureOnCompactsAffineTopologicalMackeyQuotientMeasure.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  {xi : E} (hxi : xi ≠ 0) :
  MeasureTheory.IsFiniteMeasureOnCompacts
    (LeanRidgelet.affineTopologicalMackeyQuotientMeasure
      hxi)

The transported homogeneous-space measure is finite on compact sets.

theoremdefined in LeanRidgelet/HA/AffineMackeyMeasure.lean

complete

theorem LeanRidgelet.affineTopologicalMackeyQuotient_measurePreserving.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] {xi : E}
  (hxi : xi ≠ 0) :
  MeasureTheory.MeasurePreserving
    (⇑(LeanRidgelet.affineTopologicalMackeyQuotientHomeomorphDualOrbit
        hxi))
    (LeanRidgelet.affineTopologicalMackeyQuotientMeasure hxi)
    LeanRidgelet.affineDualOrbitSubtypeMeasure

theorem LeanRidgelet.affineTopologicalMackeyQuotient_measurePreserving.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  {xi : E} (hxi : xi ≠ 0) :
  MeasureTheory.MeasurePreserving
    (⇑(LeanRidgelet.affineTopologicalMackeyQuotientHomeomorphDualOrbit
        hxi))
    (LeanRidgelet.affineTopologicalMackeyQuotientMeasure
      hxi)
    LeanRidgelet.affineDualOrbitSubtypeMeasure

The quotient-orbit homeomorphism preserves the transported homogeneous-space measure.

theoremdefined in LeanRidgelet/HA/AffineMackeyMeasure.lean

complete

theorem LeanRidgelet.affineTopologicalMackeyQuotient_symm_measurePreserving.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] {xi : E}
  (hxi : xi ≠ 0) :
  MeasureTheory.MeasurePreserving
    (⇑(LeanRidgelet.affineTopologicalMackeyQuotientHomeomorphDualOrbit
          hxi).symm)
    LeanRidgelet.affineDualOrbitSubtypeMeasure
    (LeanRidgelet.affineTopologicalMackeyQuotientMeasure hxi)

theorem LeanRidgelet.affineTopologicalMackeyQuotient_symm_measurePreserving.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  {xi : E} (hxi : xi ≠ 0) :
  MeasureTheory.MeasurePreserving
    (⇑(LeanRidgelet.affineTopologicalMackeyQuotientHomeomorphDualOrbit
          hxi).symm)
    LeanRidgelet.affineDualOrbitSubtypeMeasure
    (LeanRidgelet.affineTopologicalMackeyQuotientMeasure
      hxi)

The inverse quotient-orbit homeomorphism is also measure preserving.

defdefined in LeanRidgelet/HA/AffineMackey.lean

complete

def LeanRidgelet.affineTopologicalMackeyQuotientLpLinearIsometry.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] {xi : E}
  (hxi : xi ≠ 0) :
  ↥(MeasureTheory.Lp ℂ 2
        LeanRidgelet.affineDualOrbitSubtypeMeasure) →ₗᵢ[ℂ]
    ↥(MeasureTheory.Lp ℂ 2
        (LeanRidgelet.affineTopologicalMackeyQuotientMeasure hxi))

def LeanRidgelet.affineTopologicalMackeyQuotientLpLinearIsometry.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  {xi : E} (hxi : xi ≠ 0) :
  ↥(MeasureTheory.Lp ℂ 2
        LeanRidgelet.affineDualOrbitSubtypeMeasure) →ₗᵢ[ℂ]
    ↥(MeasureTheory.Lp ℂ 2
        (LeanRidgelet.affineTopologicalMackeyQuotientMeasure
          hxi))

Implementation after :=

:= E)) →ₗᵢ[ℂ]
      Lp ℂ 2 (affineTopologicalMackeyQuotientMeasure hxi) :=
  Lp.compMeasurePreservingₗᵢ ℂ
    (affineTopologicalMackeyQuotientHomeomorphDualOrbit hxi)
    (affineTopologicalMackeyQuotient_measurePreserving hxi)

Pullback through the quotient-orbit homeomorphism as a linear isometry from orbit `L²` to
homogeneous-space `L²`.

theoremdefined in LeanRidgelet/HA/AffineMackey.lean

complete

theorem LeanRidgelet.affineTopologicalMackeyQuotientLpLinearIsometry_apply_ae.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] {xi : E}
  (hxi : xi ≠ 0)
  (f :
    ↥(MeasureTheory.Lp ℂ 2
        LeanRidgelet.affineDualOrbitSubtypeMeasure)) :
  ↑↑((LeanRidgelet.affineTopologicalMackeyQuotientLpLinearIsometry hxi)
          f) =ᵐ[LeanRidgelet.affineTopologicalMackeyQuotientMeasure hxi]
    ↑↑f ∘
      ⇑(LeanRidgelet.affineTopologicalMackeyQuotientHomeomorphDualOrbit
          hxi)

theorem LeanRidgelet.affineTopologicalMackeyQuotientLpLinearIsometry_apply_ae.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  {xi : E} (hxi : xi ≠ 0)
  (f :
    ↥(MeasureTheory.Lp ℂ 2
        LeanRidgelet.affineDualOrbitSubtypeMeasure)) :
  ↑↑((LeanRidgelet.affineTopologicalMackeyQuotientLpLinearIsometry
            hxi)
          f) =ᵐ[LeanRidgelet.affineTopologicalMackeyQuotientMeasure
      hxi]
    ↑↑f ∘
      ⇑(LeanRidgelet.affineTopologicalMackeyQuotientHomeomorphDualOrbit
          hxi)

The homogeneous-space pullback is represented almost everywhere by composition with the
quotient-orbit homeomorphism.

theoremdefined in LeanRidgelet/HA/AffineMackey.lean

complete

theorem LeanRidgelet.affineTopologicalMackeyQuotientLpLinearIsometry_surjective.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] {xi : E}
  (hxi : xi ≠ 0) :
  Function.Surjective
    ⇑(LeanRidgelet.affineTopologicalMackeyQuotientLpLinearIsometry hxi)

theorem LeanRidgelet.affineTopologicalMackeyQuotientLpLinearIsometry_surjective.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  {xi : E} (hxi : xi ≠ 0) :
  Function.Surjective
    ⇑(LeanRidgelet.affineTopologicalMackeyQuotientLpLinearIsometry
        hxi)

Pullback through the quotient-orbit homeomorphism is onto; its inverse is pullback through the
inverse homeomorphism.

defdefined in LeanRidgelet/HA/AffineMackey.lean

complete

def LeanRidgelet.affineTopologicalMackeyQuotientLpEquiv.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E] {xi : E} (hxi : xi ≠ 0) :
  ↥(MeasureTheory.Lp ℂ 2
        LeanRidgelet.affineDualOrbitSubtypeMeasure) ≃ₗᵢ[ℂ]
    ↥(MeasureTheory.Lp ℂ 2
        (LeanRidgelet.affineTopologicalMackeyQuotientMeasure hxi))

def LeanRidgelet.affineTopologicalMackeyQuotientLpEquiv.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  {xi : E} (hxi : xi ≠ 0) :
  ↥(MeasureTheory.Lp ℂ 2
        LeanRidgelet.affineDualOrbitSubtypeMeasure) ≃ₗᵢ[ℂ]
    ↥(MeasureTheory.Lp ℂ 2
        (LeanRidgelet.affineTopologicalMackeyQuotientMeasure
          hxi))

Implementation after :=

:= E)) ≃ₗᵢ[ℂ]
      Lp ℂ 2 (affineTopologicalMackeyQuotientMeasure hxi) :=
  LinearIsometryEquiv.ofSurjective
    (affineTopologicalMackeyQuotientLpLinearIsometry hxi)
    (affineTopologicalMackeyQuotientLpLinearIsometry_surjective hxi)

The intrinsic orbit `L²` space and the homogeneous quotient `L²` space are canonically
linearly isometric.

theoremdefined in LeanRidgelet/HA/AffineMackey.lean

complete

theorem LeanRidgelet.affineTopologicalMackeyQuotientLpEquiv_apply.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] {xi : E}
  (hxi : xi ≠ 0)
  (f :
    ↥(MeasureTheory.Lp ℂ 2
        LeanRidgelet.affineDualOrbitSubtypeMeasure)) :
  (LeanRidgelet.affineTopologicalMackeyQuotientLpEquiv hxi) f =
    (LeanRidgelet.affineTopologicalMackeyQuotientLpLinearIsometry hxi) f

theorem LeanRidgelet.affineTopologicalMackeyQuotientLpEquiv_apply.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  {xi : E} (hxi : xi ≠ 0)
  (f :
    ↥(MeasureTheory.Lp ℂ 2
        LeanRidgelet.affineDualOrbitSubtypeMeasure)) :
  (LeanRidgelet.affineTopologicalMackeyQuotientLpEquiv
        hxi)
      f =
    (LeanRidgelet.affineTopologicalMackeyQuotientLpLinearIsometry
        hxi)
      f

defdefined in LeanRidgelet/HA/AffineMackey.lean

complete

def LeanRidgelet.affineTopologicalMackeyQuotientLpUnitaryRepresentation.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E] [MeasurableSpace E]
  [BorelSpace E] {xi : E} (hxi : xi ≠ 0) :
  UnitaryRepresentation (AffineEquiv.TopologicalSemidirectProduct E)
    ↥(MeasureTheory.Lp ℂ 2
        (LeanRidgelet.affineTopologicalMackeyQuotientMeasure hxi))

def LeanRidgelet.affineTopologicalMackeyQuotientLpUnitaryRepresentation.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E]
  [MeasurableSpace E] [BorelSpace E]
  {xi : E} (hxi : xi ≠ 0) :
  UnitaryRepresentation
    (AffineEquiv.TopologicalSemidirectProduct
      E)
    ↥(MeasureTheory.Lp ℂ 2
        (LeanRidgelet.affineTopologicalMackeyQuotientMeasure
          hxi))

Implementation after :=

:=
  (affineTopologicalOrbitLpUnitaryRepresentation (E := E)).conjugate
    (affineTopologicalMackeyQuotientLpEquiv hxi)

The explicit affine representation in its homogeneous-space `L²(G/H)` model.  It is
transported through the quotient-orbit homeomorphism, so no representative or measurable section
of `G/H` is chosen.

defdefined in LeanRidgelet/HA/AffineMackey.lean

complete

def LeanRidgelet.affineTopologicalMackeyQuotientIntertwiningMap.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E] [MeasurableSpace E]
  [BorelSpace E] {xi : E} (hxi : xi ≠ 0) :
  ContIntertwiningMap
    LeanRidgelet.affineTopologicalOrbitLpUnitaryRepresentation.toContRepresentation
    (LeanRidgelet.affineTopologicalMackeyQuotientLpUnitaryRepresentation
        hxi).toContRepresentation

def LeanRidgelet.affineTopologicalMackeyQuotientIntertwiningMap.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E]
  [MeasurableSpace E] [BorelSpace E]
  {xi : E} (hxi : xi ≠ 0) :
  ContIntertwiningMap
    LeanRidgelet.affineTopologicalOrbitLpUnitaryRepresentation.toContRepresentation
    (LeanRidgelet.affineTopologicalMackeyQuotientLpUnitaryRepresentation
        hxi).toContRepresentation

Implementation after :=

:= E)).toContRepresentation →ⁱL
      (affineTopologicalMackeyQuotientLpUnitaryRepresentation hxi).toContRepresentation :=
  UnitaryRepresentation.conjugateIntertwiningMap
    (affineTopologicalOrbitLpUnitaryRepresentation (E := E))
    (affineTopologicalMackeyQuotientLpEquiv hxi)

The orbit/homogeneous-space identification as a bounded intertwiner.

defdefined in LeanRidgelet/HA/AffineMackey.lean

complete

def LeanRidgelet.affineTopologicalMackeyQuotientInverseIntertwiningMap.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E] [MeasurableSpace E]
  [BorelSpace E] {xi : E} (hxi : xi ≠ 0) :
  ContIntertwiningMap
    (LeanRidgelet.affineTopologicalMackeyQuotientLpUnitaryRepresentation
        hxi).toContRepresentation
    LeanRidgelet.affineTopologicalOrbitLpUnitaryRepresentation.toContRepresentation

def LeanRidgelet.affineTopologicalMackeyQuotientInverseIntertwiningMap.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E]
  [MeasurableSpace E] [BorelSpace E]
  {xi : E} (hxi : xi ≠ 0) :
  ContIntertwiningMap
    (LeanRidgelet.affineTopologicalMackeyQuotientLpUnitaryRepresentation
        hxi).toContRepresentation
    LeanRidgelet.affineTopologicalOrbitLpUnitaryRepresentation.toContRepresentation

Implementation after :=

:= E)).toContRepresentation :=
  UnitaryRepresentation.conjugateInverseIntertwiningMap
    (affineTopologicalOrbitLpUnitaryRepresentation (E := E))
    (affineTopologicalMackeyQuotientLpEquiv hxi)

The inverse homogeneous-space/orbit identification as a bounded intertwiner.

theoremdefined in LeanRidgelet/HA/AffineMackey.lean

complete

theorem LeanRidgelet.affineTopologicalMackeyQuotientIntertwiningMap_apply.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E] [MeasurableSpace E]
  [BorelSpace E] {xi : E} (hxi : xi ≠ 0)
  (f :
    ↥(MeasureTheory.Lp ℂ 2
        LeanRidgelet.affineDualOrbitSubtypeMeasure)) :
  (LeanRidgelet.affineTopologicalMackeyQuotientIntertwiningMap hxi) f =
    (LeanRidgelet.affineTopologicalMackeyQuotientLpEquiv hxi) f

theorem LeanRidgelet.affineTopologicalMackeyQuotientIntertwiningMap_apply.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E]
  [MeasurableSpace E] [BorelSpace E]
  {xi : E} (hxi : xi ≠ 0)
  (f :
    ↥(MeasureTheory.Lp ℂ 2
        LeanRidgelet.affineDualOrbitSubtypeMeasure)) :
  (LeanRidgelet.affineTopologicalMackeyQuotientIntertwiningMap
        hxi)
      f =
    (LeanRidgelet.affineTopologicalMackeyQuotientLpEquiv
        hxi)
      f

theoremdefined in LeanRidgelet/HA/AffineMackey.lean

complete

theorem LeanRidgelet.affineTopologicalMackeyQuotientInverseIntertwiningMap_apply.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E] [MeasurableSpace E]
  [BorelSpace E] {xi : E} (hxi : xi ≠ 0)
  (f :
    ↥(MeasureTheory.Lp ℂ 2
        (LeanRidgelet.affineTopologicalMackeyQuotientMeasure hxi))) :
  (LeanRidgelet.affineTopologicalMackeyQuotientInverseIntertwiningMap
        hxi)
      f =
    (LeanRidgelet.affineTopologicalMackeyQuotientLpEquiv hxi).symm f

theorem LeanRidgelet.affineTopologicalMackeyQuotientInverseIntertwiningMap_apply.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E]
  [MeasurableSpace E] [BorelSpace E]
  {xi : E} (hxi : xi ≠ 0)
  (f :
    ↥(MeasureTheory.Lp ℂ 2
        (LeanRidgelet.affineTopologicalMackeyQuotientMeasure
          hxi))) :
  (LeanRidgelet.affineTopologicalMackeyQuotientInverseIntertwiningMap
        hxi)
      f =
    (LeanRidgelet.affineTopologicalMackeyQuotientLpEquiv
          hxi).symm
      f

theoremdefined in LeanRidgelet/HA/AffineMackey.lean

complete

theorem LeanRidgelet.affineTopologicalMackeyQuotientLpUnitaryRepresentation_isStronglyContinuous.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E] [MeasurableSpace E]
  [BorelSpace E] {xi : E} (hxi : xi ≠ 0) :
  (LeanRidgelet.affineTopologicalMackeyQuotientLpUnitaryRepresentation
      hxi).IsStronglyContinuous

theorem LeanRidgelet.affineTopologicalMackeyQuotientLpUnitaryRepresentation_isStronglyContinuous.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E]
  [MeasurableSpace E] [BorelSpace E]
  {xi : E} (hxi : xi ≠ 0) :
  (LeanRidgelet.affineTopologicalMackeyQuotientLpUnitaryRepresentation
      hxi).IsStronglyContinuous

The homogeneous-space representation is strongly continuous.

theoremdefined in LeanRidgelet/HA/AffineMackey.lean

complete

theorem LeanRidgelet.affineTopologicalMackeyQuotientLpUnitaryRepresentation_isTopologicallyIrreducible_iff.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E] [MeasurableSpace E]
  [BorelSpace E] {xi : E} (hxi : xi ≠ 0) :
  (LeanRidgelet.affineTopologicalMackeyQuotientLpUnitaryRepresentation
        hxi).IsTopologicallyIrreducible ↔
    (LeanRidgelet.affineDataLpUnitaryRepresentation
        MeasureTheory.volume).IsTopologicallyIrreducible

theorem LeanRidgelet.affineTopologicalMackeyQuotientLpUnitaryRepresentation_isTopologicallyIrreducible_iff.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E]
  [MeasurableSpace E] [BorelSpace E]
  {xi : E} (hxi : xi ≠ 0) :
  (LeanRidgelet.affineTopologicalMackeyQuotientLpUnitaryRepresentation
        hxi).IsTopologicallyIrreducible ↔
    (LeanRidgelet.affineDataLpUnitaryRepresentation
        MeasureTheory.volume).IsTopologicallyIrreducible

The homogeneous-space, orbit, Fourier, and physical affine models are topologically
irreducible simultaneously.

Definition4.3.5

Statement uses 2

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Definition 4.2.5

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used by 1✓L∃∀N

Quasi-invariance and the character twist. Injectivity of measure pushforward through the measurable orbit inclusion first transfers the ambient determinant formula to the intrinsic orbit. Equivariance of the quotient-orbit homeomorphism then gives the quotient formula both as a constant scalar multiple and as withDensity. The general quasi-invariant construction produces the untwisted quasi-regular unitary representation, whose representative is |\det L|^{1/2} f(g^{-1}q). The separately defined character phase is continuous, pointwise unimodular, satisfies the multiplier cocycle law, and restricts on translations to the frequency character. The general twisting construction bundles their product and proves its explicit a.e. action formula. The inverse transport equivalences are also identified a.e.; quasi-measure- preserving composition then proves equality with the independently transported Fourier model.

Lean code for Definition4.3.5●29 declarations

defdefined in LeanRidgelet/HA/AffineMackeyMeasure.lean

complete

def LeanRidgelet.affineTopologicalDualJacobian.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  (g : AffineEquiv.TopologicalSemidirectProduct E) : NNReal

def LeanRidgelet.affineTopologicalDualJacobian.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  (g :
    AffineEquiv.TopologicalSemidirectProduct
      E) :
  NNReal

Implementation after :=

:=
  affineDualJacobian (AffineEquiv.topologicalSemidirectProductEquiv E g) 0

The constant Radon--Nikodym density of the inverse dual action, written in the topological
semidirect-product coordinates used by Mackey theory.

theoremdefined in LeanRidgelet/HA/AffineMackeyMeasure.lean

complete

theorem LeanRidgelet.affineTopologicalDualJacobian_eq_inv.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  (g : AffineEquiv.TopologicalSemidirectProduct E) :
  LeanRidgelet.affineTopologicalDualJacobian g =
    (LeanRidgelet.affineTopologicalJacobian g)⁻¹

theorem LeanRidgelet.affineTopologicalDualJacobian_eq_inv.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  (g :
    AffineEquiv.TopologicalSemidirectProduct
      E) :
  LeanRidgelet.affineTopologicalDualJacobian
      g =
    (LeanRidgelet.affineTopologicalJacobian
        g)⁻¹

The dual density is the inverse of the data-side affine density.

theoremdefined in LeanRidgelet/HA/AffineMackeyMeasure.lean

complete

theorem LeanRidgelet.measurable_affineTopologicalDualOrbit_smul.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
  (g : AffineEquiv.TopologicalSemidirectProduct E) :
  Measurable fun eta ↦ g • eta

theorem LeanRidgelet.measurable_affineTopologicalDualOrbit_smul.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  (g :
    AffineEquiv.TopologicalSemidirectProduct
      E) :
  Measurable fun eta ↦ g • eta

Each affine group element acts measurably on the intrinsic nonzero dual orbit.

theoremdefined in LeanRidgelet/HA/AffineMackeyMeasure.lean

complete

theorem LeanRidgelet.affineDualOrbitSubtypeMeasure_map_inv_smul.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E] [MeasurableSpace E]
  [BorelSpace E] (g : AffineEquiv.TopologicalSemidirectProduct E) :
  MeasureTheory.Measure.map (fun eta ↦ g⁻¹ • eta)
      LeanRidgelet.affineDualOrbitSubtypeMeasure =
    ↑(LeanRidgelet.affineTopologicalDualJacobian g) •
      LeanRidgelet.affineDualOrbitSubtypeMeasure

theorem LeanRidgelet.affineDualOrbitSubtypeMeasure_map_inv_smul.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E]
  [MeasurableSpace E] [BorelSpace E]
  (g :
    AffineEquiv.TopologicalSemidirectProduct
      E) :
  MeasureTheory.Measure.map
      (fun eta ↦ g⁻¹ • eta)
      LeanRidgelet.affineDualOrbitSubtypeMeasure =
    ↑(LeanRidgelet.affineTopologicalDualJacobian
          g) •
      LeanRidgelet.affineDualOrbitSubtypeMeasure

The intrinsic orbit measure is strongly quasi-invariant under the dual affine action, with
the same constant density as ambient Lebesgue measure.

theoremdefined in LeanRidgelet/HA/AffineMackeyMeasure.lean

complete

theorem LeanRidgelet.affineDualOrbitSubtype_quasiMeasurePreserving.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E] [MeasurableSpace E]
  [BorelSpace E] (g : AffineEquiv.TopologicalSemidirectProduct E) :
  MeasureTheory.Measure.QuasiMeasurePreserving (fun eta ↦ g⁻¹ • eta)
    LeanRidgelet.affineDualOrbitSubtypeMeasure
    LeanRidgelet.affineDualOrbitSubtypeMeasure

theorem LeanRidgelet.affineDualOrbitSubtype_quasiMeasurePreserving.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E]
  [MeasurableSpace E] [BorelSpace E]
  (g :
    AffineEquiv.TopologicalSemidirectProduct
      E) :
  MeasureTheory.Measure.QuasiMeasurePreserving
    (fun eta ↦ g⁻¹ • eta)
    LeanRidgelet.affineDualOrbitSubtypeMeasure
    LeanRidgelet.affineDualOrbitSubtypeMeasure

Inverse dual translation is quasi-measure-preserving on the intrinsic nonzero orbit.

defdefined in LeanRidgelet/HA/AffineMackeyMeasure.lean

complete

def LeanRidgelet.affineTopologicalMackeyQuotientJacobian.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {xi : E}
  (g : AffineEquiv.TopologicalSemidirectProduct E) :
  LeanRidgelet.AffineTopologicalMackeyQuotient xi → NNReal

def LeanRidgelet.affineTopologicalMackeyQuotientJacobian.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {xi : E}
  (g :
    AffineEquiv.TopologicalSemidirectProduct
      E) :
  LeanRidgelet.AffineTopologicalMackeyQuotient
      xi →
    NNReal

Implementation after :=

:=
  affineTopologicalDualJacobian g

The Radon--Nikodym cocycle for the transported measure on the affine homogeneous space.  It
is constant on the quotient because the dual action has constant determinant density.

theoremdefined in LeanRidgelet/HA/AffineMackeyMeasure.lean

complete

theorem LeanRidgelet.measurable_affineTopologicalMackeyQuotient_smul.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {xi : E}
  (g : AffineEquiv.TopologicalSemidirectProduct E) :
  Measurable fun q ↦ g • q

theorem LeanRidgelet.measurable_affineTopologicalMackeyQuotient_smul.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {xi : E}
  (g :
    AffineEquiv.TopologicalSemidirectProduct
      E) :
  Measurable fun q ↦ g • q

Left translation on the affine homogeneous quotient is measurable.

theoremdefined in LeanRidgelet/HA/AffineMackeyMeasure.lean

complete

theorem LeanRidgelet.affineTopologicalMackeyQuotientMeasure_map_inv_smul.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E] [MeasurableSpace E]
  [BorelSpace E] {xi : E} (hxi : xi ≠ 0)
  (g : AffineEquiv.TopologicalSemidirectProduct E) :
  MeasureTheory.Measure.map (fun q ↦ g⁻¹ • q)
      (LeanRidgelet.affineTopologicalMackeyQuotientMeasure hxi) =
    ↑(LeanRidgelet.affineTopologicalDualJacobian g) •
      LeanRidgelet.affineTopologicalMackeyQuotientMeasure hxi

theorem LeanRidgelet.affineTopologicalMackeyQuotientMeasure_map_inv_smul.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E]
  [MeasurableSpace E] [BorelSpace E]
  {xi : E} (hxi : xi ≠ 0)
  (g :
    AffineEquiv.TopologicalSemidirectProduct
      E) :
  MeasureTheory.Measure.map
      (fun q ↦ g⁻¹ • q)
      (LeanRidgelet.affineTopologicalMackeyQuotientMeasure
        hxi) =
    ↑(LeanRidgelet.affineTopologicalDualJacobian
          g) •
      LeanRidgelet.affineTopologicalMackeyQuotientMeasure
        hxi

The quotient-orbit homeomorphism transports the constant-density quasi-invariance of the
intrinsic orbit measure to the homogeneous space.

theoremdefined in LeanRidgelet/HA/AffineMackeyMeasure.lean

complete

theorem LeanRidgelet.affineTopologicalMackeyQuotientJacobian_measurable.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {xi : E}
  (g : AffineEquiv.TopologicalSemidirectProduct E) :
  Measurable (LeanRidgelet.affineTopologicalMackeyQuotientJacobian g)

theorem LeanRidgelet.affineTopologicalMackeyQuotientJacobian_measurable.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {xi : E}
  (g :
    AffineEquiv.TopologicalSemidirectProduct
      E) :
  Measurable
    (LeanRidgelet.affineTopologicalMackeyQuotientJacobian
      g)

The homogeneous-space Jacobian is measurable.

theoremdefined in LeanRidgelet/HA/AffineMackeyMeasure.lean

complete

theorem LeanRidgelet.affineTopologicalMackeyQuotientJacobian_ne_zero.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {xi : E}
  (g : AffineEquiv.TopologicalSemidirectProduct E)
  (q : LeanRidgelet.AffineTopologicalMackeyQuotient xi) :
  LeanRidgelet.affineTopologicalMackeyQuotientJacobian g q ≠ 0

theorem LeanRidgelet.affineTopologicalMackeyQuotientJacobian_ne_zero.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {xi : E}
  (g :
    AffineEquiv.TopologicalSemidirectProduct
      E)
  (q :
    LeanRidgelet.AffineTopologicalMackeyQuotient
      xi) :
  LeanRidgelet.affineTopologicalMackeyQuotientJacobian
      g q ≠
    0

The homogeneous-space Jacobian is everywhere nonzero.

theoremdefined in LeanRidgelet/HA/AffineMackeyMeasure.lean

complete

theorem LeanRidgelet.affineTopologicalMackeyQuotientJacobian_one.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {xi : E}
  (q : LeanRidgelet.AffineTopologicalMackeyQuotient xi) :
  LeanRidgelet.affineTopologicalMackeyQuotientJacobian 1 q = 1

theorem LeanRidgelet.affineTopologicalMackeyQuotientJacobian_one.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {xi : E}
  (q :
    LeanRidgelet.AffineTopologicalMackeyQuotient
      xi) :
  LeanRidgelet.affineTopologicalMackeyQuotientJacobian
      1 q =
    1

The homogeneous-space Jacobian at the identity is one.

theoremdefined in LeanRidgelet/HA/AffineMackeyMeasure.lean

complete

theorem LeanRidgelet.affineTopologicalMackeyQuotientJacobian_cocycle.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {xi : E}
  (g h : AffineEquiv.TopologicalSemidirectProduct E)
  (q : LeanRidgelet.AffineTopologicalMackeyQuotient xi) :
  LeanRidgelet.affineTopologicalMackeyQuotientJacobian (g * h) q =
    LeanRidgelet.affineTopologicalMackeyQuotientJacobian g (h • q) *
      LeanRidgelet.affineTopologicalMackeyQuotientJacobian h q

theorem LeanRidgelet.affineTopologicalMackeyQuotientJacobian_cocycle.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {xi : E}
  (g h :
    AffineEquiv.TopologicalSemidirectProduct
      E)
  (q :
    LeanRidgelet.AffineTopologicalMackeyQuotient
      xi) :
  LeanRidgelet.affineTopologicalMackeyQuotientJacobian
      (g * h) q =
    LeanRidgelet.affineTopologicalMackeyQuotientJacobian
        g (h • q) *
      LeanRidgelet.affineTopologicalMackeyQuotientJacobian
        h q

The homogeneous-space density satisfies the Radon--Nikodym cocycle law.

theoremdefined in LeanRidgelet/HA/AffineMackeyMeasure.lean

complete

theorem LeanRidgelet.affineTopologicalMackeyQuotientMeasure_map_eq_withDensity.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E] [MeasurableSpace E]
  [BorelSpace E] {xi : E} (hxi : xi ≠ 0)
  (g : AffineEquiv.TopologicalSemidirectProduct E) :
  MeasureTheory.Measure.map (fun q ↦ g⁻¹ • q)
      (LeanRidgelet.affineTopologicalMackeyQuotientMeasure hxi) =
    (LeanRidgelet.affineTopologicalMackeyQuotientMeasure
          hxi).withDensity
      fun q ↦
      ↑(LeanRidgelet.affineTopologicalMackeyQuotientJacobian g q)

theorem LeanRidgelet.affineTopologicalMackeyQuotientMeasure_map_eq_withDensity.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E]
  [MeasurableSpace E] [BorelSpace E]
  {xi : E} (hxi : xi ≠ 0)
  (g :
    AffineEquiv.TopologicalSemidirectProduct
      E) :
  MeasureTheory.Measure.map
      (fun q ↦ g⁻¹ • q)
      (LeanRidgelet.affineTopologicalMackeyQuotientMeasure
        hxi) =
    (LeanRidgelet.affineTopologicalMackeyQuotientMeasure
          hxi).withDensity
      fun q ↦
      ↑(LeanRidgelet.affineTopologicalMackeyQuotientJacobian
          g q)

Strong quasi-invariance of the homogeneous-space measure, in the `withDensity` form consumed
by the general quasi-invariant `L²` construction.

theoremdefined in LeanRidgelet/HA/AffineMackeyMeasure.lean

complete

theorem LeanRidgelet.affineTopologicalMackeyQuotientRadonNikodymWeight.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {xi : E}
  (g : AffineEquiv.TopologicalSemidirectProduct E)
  (q : LeanRidgelet.AffineTopologicalMackeyQuotient xi) :
  LeanRidgelet.radonNikodymWeight
      LeanRidgelet.affineTopologicalMackeyQuotientJacobian g q =
    ↑↑(NNReal.sqrt (LeanRidgelet.affineTopologicalJacobian g))

theorem LeanRidgelet.affineTopologicalMackeyQuotientRadonNikodymWeight.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {xi : E}
  (g :
    AffineEquiv.TopologicalSemidirectProduct
      E)
  (q :
    LeanRidgelet.AffineTopologicalMackeyQuotient
      xi) :
  LeanRidgelet.radonNikodymWeight
      LeanRidgelet.affineTopologicalMackeyQuotientJacobian
      g q =
    ↑↑(NNReal.sqrt
          (LeanRidgelet.affineTopologicalJacobian
            g))

Folland's Radon--Nikodym correction on the homogeneous quotient is the positive square root
of the data-side affine determinant.

defdefined in LeanRidgelet/HA/AffineMackeyInduction.lean

complete

def LeanRidgelet.affineTopologicalMackeyQuotientPhase.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  {xi : E} (hxi : xi ≠ 0)
  (g : AffineEquiv.TopologicalSemidirectProduct E)
  (q : LeanRidgelet.AffineTopologicalMackeyQuotient xi) : ℂ

def LeanRidgelet.affineTopologicalMackeyQuotientPhase.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {xi : E}
  (hxi : xi ≠ 0)
  (g :
    AffineEquiv.TopologicalSemidirectProduct
      E)
  (q :
    LeanRidgelet.AffineTopologicalMackeyQuotient
      xi) :
  ℂ

Implementation after :=

:=
  affineFrequencyPhase (AffineEquiv.topologicalSemidirectProductEquiv E g)
    (affineTopologicalMackeyQuotientHomeomorphDualOrbit hxi q).1

The Mackey character phase on the affine homogeneous quotient, obtained from the frequency
attached to a coset by the quotient-orbit homeomorphism.

theoremdefined in LeanRidgelet/HA/AffineMackeyInduction.lean

complete

theorem LeanRidgelet.affineTopologicalMackeyQuotientPhase_cocycle.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {xi : E} (hxi : xi ≠ 0)
  (g h : AffineEquiv.TopologicalSemidirectProduct E)
  (q : LeanRidgelet.AffineTopologicalMackeyQuotient xi) :
  LeanRidgelet.affineTopologicalMackeyQuotientPhase hxi (g * h) q =
    LeanRidgelet.affineTopologicalMackeyQuotientPhase hxi g q *
      LeanRidgelet.affineTopologicalMackeyQuotientPhase hxi h (g⁻¹ • q)

theorem LeanRidgelet.affineTopologicalMackeyQuotientPhase_cocycle.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {xi : E}
  (hxi : xi ≠ 0)
  (g h :
    AffineEquiv.TopologicalSemidirectProduct
      E)
  (q :
    LeanRidgelet.AffineTopologicalMackeyQuotient
      xi) :
  LeanRidgelet.affineTopologicalMackeyQuotientPhase
      hxi (g * h) q =
    LeanRidgelet.affineTopologicalMackeyQuotientPhase
        hxi g q *
      LeanRidgelet.affineTopologicalMackeyQuotientPhase
        hxi h (g⁻¹ • q)

The quotient phase is a multiplier cocycle for inverse left translation.

theoremdefined in LeanRidgelet/HA/AffineMackeyInduction.lean

complete

theorem LeanRidgelet.affineTopologicalMackeyQuotientPhase_norm_one.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {xi : E} (hxi : xi ≠ 0)
  (g : AffineEquiv.TopologicalSemidirectProduct E)
  (q : LeanRidgelet.AffineTopologicalMackeyQuotient xi) :
  ‖LeanRidgelet.affineTopologicalMackeyQuotientPhase hxi g q‖ = 1

theorem LeanRidgelet.affineTopologicalMackeyQuotientPhase_norm_one.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {xi : E}
  (hxi : xi ≠ 0)
  (g :
    AffineEquiv.TopologicalSemidirectProduct
      E)
  (q :
    LeanRidgelet.AffineTopologicalMackeyQuotient
      xi) :
  ‖LeanRidgelet.affineTopologicalMackeyQuotientPhase
        hxi g q‖ =
    1

The Mackey character phase is pointwise unimodular.

theoremdefined in LeanRidgelet/HA/AffineMackeyInduction.lean

complete

theorem LeanRidgelet.continuous_affineTopologicalMackeyQuotientPhase.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {xi : E} (hxi : xi ≠ 0)
  (g : AffineEquiv.TopologicalSemidirectProduct E) :
  Continuous (LeanRidgelet.affineTopologicalMackeyQuotientPhase hxi g)

theorem LeanRidgelet.continuous_affineTopologicalMackeyQuotientPhase.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {xi : E}
  (hxi : xi ≠ 0)
  (g :
    AffineEquiv.TopologicalSemidirectProduct
      E) :
  Continuous
    (LeanRidgelet.affineTopologicalMackeyQuotientPhase
      hxi g)

The Mackey character phase is continuous in the homogeneous-space variable.

theoremdefined in LeanRidgelet/HA/AffineMackeyInduction.lean

complete

theorem LeanRidgelet.affineTopologicalMackeyQuotientPhase_translation.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {xi : E} (hxi : xi ≠ 0) (b : E)
  (q : LeanRidgelet.AffineTopologicalMackeyQuotient xi) :
  LeanRidgelet.affineTopologicalMackeyQuotientPhase hxi
      ((AffineEquiv.topologicalSemidirectProductEquiv E).symm
        (LeanRidgelet.affineTranslation b))
      q =
    ↑((LeanRidgelet.affineTranslationCharacter
          ↑((LeanRidgelet.affineTopologicalMackeyQuotientHomeomorphDualOrbit
                hxi)
              q))
        b)

theorem LeanRidgelet.affineTopologicalMackeyQuotientPhase_translation.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {xi : E}
  (hxi : xi ≠ 0) (b : E)
  (q :
    LeanRidgelet.AffineTopologicalMackeyQuotient
      xi) :
  LeanRidgelet.affineTopologicalMackeyQuotientPhase
      hxi
      ((AffineEquiv.topologicalSemidirectProductEquiv
            E).symm
        (LeanRidgelet.affineTranslation
          b))
      q =
    ↑((LeanRidgelet.affineTranslationCharacter
          ↑((LeanRidgelet.affineTopologicalMackeyQuotientHomeomorphDualOrbit
                hxi)
              q))
        b)

On translations, the quotient phase is exactly the Mackey translation character at the
frequency represented by the coset.

defdefined in LeanRidgelet/HA/AffineMackeyInduction.lean

complete

def LeanRidgelet.affineTopologicalMackeyQuotientQuasiRegularLpUnitaryRepresentation.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E] [MeasurableSpace E]
  [BorelSpace E] {xi : E} (hxi : xi ≠ 0) :
  UnitaryRepresentation (AffineEquiv.TopologicalSemidirectProduct E)
    ↥(MeasureTheory.Lp ℂ 2
        (LeanRidgelet.affineTopologicalMackeyQuotientMeasure hxi))

def LeanRidgelet.affineTopologicalMackeyQuotientQuasiRegularLpUnitaryRepresentation.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E]
  [MeasurableSpace E] [BorelSpace E]
  {xi : E} (hxi : xi ≠ 0) :
  UnitaryRepresentation
    (AffineEquiv.TopologicalSemidirectProduct
      E)
    ↥(MeasureTheory.Lp ℂ 2
        (LeanRidgelet.affineTopologicalMackeyQuotientMeasure
          hxi))

Implementation after :=

:=
  quasiInvariantLpUnitaryRepresentation
    (affineTopologicalMackeyQuotientJacobian (xi := xi))
    measurable_affineTopologicalMackeyQuotient_smul
    (affineTopologicalMackeyQuotientMeasure_map_eq_withDensity hxi)
    affineTopologicalMackeyQuotientJacobian_measurable
    affineTopologicalMackeyQuotientJacobian_ne_zero
    affineTopologicalMackeyQuotientJacobian_one
    affineTopologicalMackeyQuotientJacobian_cocycle

The untwisted quasi-regular unitary representation on the affine homogeneous quotient.  This
is the measure-theoretic base of the Mackey model; the inducing character supplies an additional
unit-modulus phase in the representation transported from the frequency orbit below.

theoremdefined in LeanRidgelet/HA/AffineMackeyInduction.lean

complete

theorem LeanRidgelet.affineTopologicalMackeyQuotientQuasiRegularLpUnitaryRepresentation_apply_ae.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E] [MeasurableSpace E]
  [BorelSpace E] {xi : E} (hxi : xi ≠ 0)
  (g : AffineEquiv.TopologicalSemidirectProduct E)
  (f :
    ↥(MeasureTheory.Lp ℂ 2
        (LeanRidgelet.affineTopologicalMackeyQuotientMeasure hxi))) :
  ↑↑(↑((LeanRidgelet.affineTopologicalMackeyQuotientQuasiRegularLpUnitaryRepresentation
                hxi)
              g)
          f) =ᵐ[LeanRidgelet.affineTopologicalMackeyQuotientMeasure hxi]
    LeanRidgelet.quasiRegularAction
      (LeanRidgelet.radonNikodymWeight
        LeanRidgelet.affineTopologicalMackeyQuotientJacobian)
      g fun q ↦ ↑↑f q

theorem LeanRidgelet.affineTopologicalMackeyQuotientQuasiRegularLpUnitaryRepresentation_apply_ae.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E]
  [MeasurableSpace E] [BorelSpace E]
  {xi : E} (hxi : xi ≠ 0)
  (g :
    AffineEquiv.TopologicalSemidirectProduct
      E)
  (f :
    ↥(MeasureTheory.Lp ℂ 2
        (LeanRidgelet.affineTopologicalMackeyQuotientMeasure
          hxi))) :
  ↑↑(↑((LeanRidgelet.affineTopologicalMackeyQuotientQuasiRegularLpUnitaryRepresentation
                hxi)
              g)
          f) =ᵐ[LeanRidgelet.affineTopologicalMackeyQuotientMeasure
      hxi]
    LeanRidgelet.quasiRegularAction
      (LeanRidgelet.radonNikodymWeight
        LeanRidgelet.affineTopologicalMackeyQuotientJacobian)
      g fun q ↦ ↑↑f q

The quotient quasi-regular representation has Folland's determinant-corrected pullback as an
almost-everywhere representative.

theoremdefined in LeanRidgelet/HA/AffineMackeyInduction.lean

complete

theorem LeanRidgelet.affineTopologicalMackeyQuotientQuasiRegularLpUnitaryRepresentation_apply_ae_explicit.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E] [MeasurableSpace E]
  [BorelSpace E] {xi : E} (hxi : xi ≠ 0)
  (g : AffineEquiv.TopologicalSemidirectProduct E)
  (f :
    ↥(MeasureTheory.Lp ℂ 2
        (LeanRidgelet.affineTopologicalMackeyQuotientMeasure hxi))) :
  ↑↑(↑((LeanRidgelet.affineTopologicalMackeyQuotientQuasiRegularLpUnitaryRepresentation
                hxi)
              g)
          f) =ᵐ[LeanRidgelet.affineTopologicalMackeyQuotientMeasure hxi]
    fun q ↦
    ↑↑(NNReal.sqrt (LeanRidgelet.affineTopologicalJacobian g)) •
      ↑↑f (g⁻¹ • q)

theorem LeanRidgelet.affineTopologicalMackeyQuotientQuasiRegularLpUnitaryRepresentation_apply_ae_explicit.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E]
  [MeasurableSpace E] [BorelSpace E]
  {xi : E} (hxi : xi ≠ 0)
  (g :
    AffineEquiv.TopologicalSemidirectProduct
      E)
  (f :
    ↥(MeasureTheory.Lp ℂ 2
        (LeanRidgelet.affineTopologicalMackeyQuotientMeasure
          hxi))) :
  ↑↑(↑((LeanRidgelet.affineTopologicalMackeyQuotientQuasiRegularLpUnitaryRepresentation
                hxi)
              g)
          f) =ᵐ[LeanRidgelet.affineTopologicalMackeyQuotientMeasure
      hxi]
    fun q ↦
    ↑↑(NNReal.sqrt
            (LeanRidgelet.affineTopologicalJacobian
              g)) •
      ↑↑f (g⁻¹ • q)

Explicitly, the quotient quasi-regular action is pullback by inverse left translation times
the square root of the affine determinant.

defdefined in LeanRidgelet/HA/AffineMackeyInduction.lean

complete

def LeanRidgelet.affineTopologicalMackeyQuotientCharacterTwistedLpUnitaryRepresentation.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E] [MeasurableSpace E]
  [BorelSpace E] {xi : E} (hxi : xi ≠ 0) :
  UnitaryRepresentation (AffineEquiv.TopologicalSemidirectProduct E)
    ↥(MeasureTheory.Lp ℂ 2
        (LeanRidgelet.affineTopologicalMackeyQuotientMeasure hxi))

def LeanRidgelet.affineTopologicalMackeyQuotientCharacterTwistedLpUnitaryRepresentation.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E]
  [MeasurableSpace E] [BorelSpace E]
  {xi : E} (hxi : xi ≠ 0) :
  UnitaryRepresentation
    (AffineEquiv.TopologicalSemidirectProduct
      E)
    ↥(MeasureTheory.Lp ℂ 2
        (LeanRidgelet.affineTopologicalMackeyQuotientMeasure
          hxi))

Implementation after :=

:=
  twistedQuasiInvariantLpUnitaryRepresentation
    (affineTopologicalMackeyQuotientJacobian (xi := xi))
    measurable_affineTopologicalMackeyQuotient_smul
    (affineTopologicalMackeyQuotientMeasure_map_eq_withDensity hxi)
    affineTopologicalMackeyQuotientJacobian_measurable
    affineTopologicalMackeyQuotientJacobian_ne_zero
    affineTopologicalMackeyQuotientJacobian_one
    affineTopologicalMackeyQuotientJacobian_cocycle
    (affineTopologicalMackeyQuotientPhase hxi)
    (fun g ↦ (continuous_affineTopologicalMackeyQuotientPhase hxi g).aestronglyMeasurable)
    (fun g ↦ Filter.Eventually.of_forall
      (affineTopologicalMackeyQuotientPhase_norm_one hxi g))
    (affineTopologicalMackeyQuotientPhase_one hxi)
    (affineTopologicalMackeyQuotientPhase_cocycle hxi)

The character-twisted quasi-regular representation on `L²(G/H)`.  Its two factors are
constructed independently: the quotient measure supplies the Radon--Nikodym correction, while
the Mackey translation character supplies the unimodular multiplier cocycle.

theoremdefined in LeanRidgelet/HA/AffineMackeyInduction.lean

complete

theorem LeanRidgelet.affineTopologicalMackeyQuotientCharacterTwistedLpUnitaryRepresentation_apply_ae_explicit.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E] [MeasurableSpace E]
  [BorelSpace E] {xi : E} (hxi : xi ≠ 0)
  (g : AffineEquiv.TopologicalSemidirectProduct E)
  (f :
    ↥(MeasureTheory.Lp ℂ 2
        (LeanRidgelet.affineTopologicalMackeyQuotientMeasure hxi))) :
  ↑↑(↑((LeanRidgelet.affineTopologicalMackeyQuotientCharacterTwistedLpUnitaryRepresentation
                hxi)
              g)
          f) =ᵐ[LeanRidgelet.affineTopologicalMackeyQuotientMeasure hxi]
    fun q ↦
    LeanRidgelet.affineTopologicalMackeyQuotientPhase hxi g q *
      (↑↑(NNReal.sqrt (LeanRidgelet.affineTopologicalJacobian g)) *
        ↑↑f (g⁻¹ • q))

theorem LeanRidgelet.affineTopologicalMackeyQuotientCharacterTwistedLpUnitaryRepresentation_apply_ae_explicit.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E]
  [MeasurableSpace E] [BorelSpace E]
  {xi : E} (hxi : xi ≠ 0)
  (g :
    AffineEquiv.TopologicalSemidirectProduct
      E)
  (f :
    ↥(MeasureTheory.Lp ℂ 2
        (LeanRidgelet.affineTopologicalMackeyQuotientMeasure
          hxi))) :
  ↑↑(↑((LeanRidgelet.affineTopologicalMackeyQuotientCharacterTwistedLpUnitaryRepresentation
                hxi)
              g)
          f) =ᵐ[LeanRidgelet.affineTopologicalMackeyQuotientMeasure
      hxi]
    fun q ↦
    LeanRidgelet.affineTopologicalMackeyQuotientPhase
        hxi g q *
      (↑↑(NNReal.sqrt
              (LeanRidgelet.affineTopologicalJacobian
                g)) *
        ↑↑f (g⁻¹ • q))

The character-twisted quotient representation acts by the Mackey phase times the positive
square root of the affine determinant and inverse left translation.

theoremdefined in LeanRidgelet/HA/AffineMackeyMeasure.lean

complete

theorem LeanRidgelet.affineDualOrbitSubtypeLpEquiv_symm_apply_ae.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E] [MeasurableSpace E]
  [BorelSpace E]
  (f :
    ↥(MeasureTheory.Lp ℂ 2
        LeanRidgelet.affineDualOrbitSubtypeMeasure)) :
  (fun eta ↦
      ↑↑(LeanRidgelet.affineDualOrbitSubtypeLpEquiv.symm f)
        ↑eta) =ᵐ[LeanRidgelet.affineDualOrbitSubtypeMeasure]
    ↑↑f

theorem LeanRidgelet.affineDualOrbitSubtypeLpEquiv_symm_apply_ae.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E]
  [MeasurableSpace E] [BorelSpace E]
  (f :
    ↥(MeasureTheory.Lp ℂ 2
        LeanRidgelet.affineDualOrbitSubtypeMeasure)) :
  (fun eta ↦
      ↑↑(LeanRidgelet.affineDualOrbitSubtypeLpEquiv.symm
              f)
        ↑eta) =ᵐ[LeanRidgelet.affineDualOrbitSubtypeMeasure]
    ↑↑f

The inverse intrinsic-orbit equivalence agrees almost everywhere with the original full-space
class on the conull nonzero subtype.

theoremdefined in LeanRidgelet/HA/AffineMackey.lean

complete

theorem LeanRidgelet.affineTopologicalMackeyQuotientLpEquiv_symm_apply_ae.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] {xi : E}
  (hxi : xi ≠ 0)
  (f :
    ↥(MeasureTheory.Lp ℂ 2
        (LeanRidgelet.affineTopologicalMackeyQuotientMeasure hxi))) :
  ↑↑((LeanRidgelet.affineTopologicalMackeyQuotientLpEquiv hxi).symm
          f) =ᵐ[LeanRidgelet.affineDualOrbitSubtypeMeasure]
    ↑↑f ∘
      ⇑(LeanRidgelet.affineTopologicalMackeyQuotientHomeomorphDualOrbit
            hxi).symm

theorem LeanRidgelet.affineTopologicalMackeyQuotientLpEquiv_symm_apply_ae.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  {xi : E} (hxi : xi ≠ 0)
  (f :
    ↥(MeasureTheory.Lp ℂ 2
        (LeanRidgelet.affineTopologicalMackeyQuotientMeasure
          hxi))) :
  ↑↑((LeanRidgelet.affineTopologicalMackeyQuotientLpEquiv
              hxi).symm
          f) =ᵐ[LeanRidgelet.affineDualOrbitSubtypeMeasure]
    ↑↑f ∘
      ⇑(LeanRidgelet.affineTopologicalMackeyQuotientHomeomorphDualOrbit
            hxi).symm

The inverse homogeneous-space equivalence is represented by pullback through the inverse
quotient-orbit homeomorphism.

theoremdefined in LeanRidgelet/HA/AffineMackey.lean

complete

theorem LeanRidgelet.affineTopologicalOrbitLpUnitaryRepresentation_apply_ae.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E] [MeasurableSpace E]
  [BorelSpace E] (g : AffineEquiv.TopologicalSemidirectProduct E)
  (f :
    ↥(MeasureTheory.Lp ℂ 2
        LeanRidgelet.affineDualOrbitSubtypeMeasure)) :
  ↑↑(↑(LeanRidgelet.affineTopologicalOrbitLpUnitaryRepresentation g)
          f) =ᵐ[LeanRidgelet.affineDualOrbitSubtypeMeasure]
    fun eta ↦
    LeanRidgelet.affineFrequencyPhase
        ((AffineEquiv.topologicalSemidirectProductEquiv E) g) ↑eta *
      (↑↑(NNReal.sqrt (LeanRidgelet.affineTopologicalJacobian g)) *
        ↑↑f (g⁻¹ • eta))

theorem LeanRidgelet.affineTopologicalOrbitLpUnitaryRepresentation_apply_ae.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E]
  [MeasurableSpace E] [BorelSpace E]
  (g :
    AffineEquiv.TopologicalSemidirectProduct
      E)
  (f :
    ↥(MeasureTheory.Lp ℂ 2
        LeanRidgelet.affineDualOrbitSubtypeMeasure)) :
  ↑↑(↑(LeanRidgelet.affineTopologicalOrbitLpUnitaryRepresentation
              g)
          f) =ᵐ[LeanRidgelet.affineDualOrbitSubtypeMeasure]
    fun eta ↦
    LeanRidgelet.affineFrequencyPhase
        ((AffineEquiv.topologicalSemidirectProductEquiv
            E)
          g)
        ↑eta *
      (↑↑(NNReal.sqrt
              (LeanRidgelet.affineTopologicalJacobian
                g)) *
        ↑↑f (g⁻¹ • eta))

The transported intrinsic-orbit representation retains the explicit
character-times-determinant-corrected dual pullback formula.

theoremdefined in LeanRidgelet/HA/AffineMackey.lean

complete

theorem LeanRidgelet.affineTopologicalMackeyQuotientLpUnitaryRepresentation_apply_ae.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E] [MeasurableSpace E]
  [BorelSpace E] {xi : E} (hxi : xi ≠ 0)
  (g : AffineEquiv.TopologicalSemidirectProduct E)
  (f :
    ↥(MeasureTheory.Lp ℂ 2
        (LeanRidgelet.affineTopologicalMackeyQuotientMeasure hxi))) :
  ↑↑(↑((LeanRidgelet.affineTopologicalMackeyQuotientLpUnitaryRepresentation
                hxi)
              g)
          f) =ᵐ[LeanRidgelet.affineTopologicalMackeyQuotientMeasure hxi]
    fun q ↦
    LeanRidgelet.affineTopologicalMackeyQuotientPhase hxi g q *
      (↑↑(NNReal.sqrt (LeanRidgelet.affineTopologicalJacobian g)) *
        ↑↑f (g⁻¹ • q))

theorem LeanRidgelet.affineTopologicalMackeyQuotientLpUnitaryRepresentation_apply_ae.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E]
  [MeasurableSpace E] [BorelSpace E]
  {xi : E} (hxi : xi ≠ 0)
  (g :
    AffineEquiv.TopologicalSemidirectProduct
      E)
  (f :
    ↥(MeasureTheory.Lp ℂ 2
        (LeanRidgelet.affineTopologicalMackeyQuotientMeasure
          hxi))) :
  ↑↑(↑((LeanRidgelet.affineTopologicalMackeyQuotientLpUnitaryRepresentation
                hxi)
              g)
          f) =ᵐ[LeanRidgelet.affineTopologicalMackeyQuotientMeasure
      hxi]
    fun q ↦
    LeanRidgelet.affineTopologicalMackeyQuotientPhase
        hxi g q *
      (↑↑(NNReal.sqrt
              (LeanRidgelet.affineTopologicalJacobian
                g)) *
        ↑↑f (g⁻¹ • q))

The independently transported homogeneous-space representation has the same explicit action
as the character-twisted quasi-regular construction.

theoremdefined in LeanRidgelet/HA/AffineMackey.lean

complete

theorem LeanRidgelet.affineTopologicalMackeyQuotientCharacterTwistedLpUnitaryRepresentation_eq_transported.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E] [MeasurableSpace E]
  [BorelSpace E] {xi : E} (hxi : xi ≠ 0) :
  LeanRidgelet.affineTopologicalMackeyQuotientCharacterTwistedLpUnitaryRepresentation
      hxi =
    LeanRidgelet.affineTopologicalMackeyQuotientLpUnitaryRepresentation
      hxi

theorem LeanRidgelet.affineTopologicalMackeyQuotientCharacterTwistedLpUnitaryRepresentation_eq_transported.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E]
  [MeasurableSpace E] [BorelSpace E]
  {xi : E} (hxi : xi ≠ 0) :
  LeanRidgelet.affineTopologicalMackeyQuotientCharacterTwistedLpUnitaryRepresentation
      hxi =
    LeanRidgelet.affineTopologicalMackeyQuotientLpUnitaryRepresentation
      hxi

The character-twisted quasi-regular representation is exactly the homogeneous-space model
obtained independently by transporting the explicit Fourier representation.

Theorem4.3.6

Statement uses 8

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used by 1✓L∃∀N

Normalized-section unitary induction. Since the inducing subgroup contains every translation, the choice-based quotient representative can be normalized by discarding its translation coordinate without changing its coset. This gives a right inverse s:G/H\to G and hence the reentry cocycle s(q)^{-1}g s(g^{-1}q) in the inducing subgroup. Applying the inducing character to this cocycle is proved, by the adjoint identity for the linear coordinate of s(q), to equal the independently constructed Fourier phase. The resulting quasi-invariant L² representation is therefore exactly both the character-twisted quotient model and the transported Fourier model; it is strongly continuous, and its irreducibility is equivalent to Theorem 2.5. No measurability of the raw choice function is assumed: regularity is obtained from the proved phase equality. Measurable-set indicator operators form its canonical projection family and satisfy P_s\pi(g)=\pi(g)P_{g^{-1}s}. Pure translations fix the quotient and have unit Jacobian, so their restriction is proved a.e. to be multiplication by the orbit-frequency character. The quotient coordinate is a measurable embedding into frequency space, and the translation restriction is identified with the general bundled Fourier-character multiplier. The derived affine Theorem-4.44 step is source-level complete: finite-character density in L^2 of an arbitrary finite measure is proved in the Mathlib candidate layer by characteristic-function uniqueness. The spectral-subspace and self-adjoint-decomposition remainder of Theorem 6.28 is proved as well. Compactly supported continuous scalar sections are bundled into L²; their identity-coset values define the extracted closed inducing fiber. Haar-integrated smoothing is defined for this affine representation. A sequence of nonnegative compactly supported Haar probability kernels is constructed with supports shrinking to the identity; the smoothed vectors stay in the invariant closed subspace and converge in L² to the input. The compact-kernel group-convolution formula is now proved continuous in the Mathlib candidate layer by reduction to additive convolution on the opposite group. Once a smoothed vector has a continuous quotient representative, the general compact-multiplier API keeps its Urysohn cutoffs in the same subspace, and a diagonal sequence proves regular-section density. The general theorem identifying a Bochner integral in scalar L² with a family of pointwise representatives integrable over the finite-measure slices is also proved, and applying it to the measurable induced-model lift finishes Lemma 6.29: the smoothed vector agrees almost everywhere with the pointwise convolution, which is continuous. Indicator and representation stability, as well as the classification of closed complex subspaces of the one-dimensional fiber, are proved. The extreme-subspace part of Lemma 6.30 is now proved as well.

Lean code for Theorem4.3.6●59 declarations

defdefined in LeanRidgelet/HA/AffineMackeyInduction.lean

complete

def LeanRidgelet.affineTopologicalMackeySection.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  {xi : E} (q : LeanRidgelet.AffineTopologicalMackeyQuotient xi) :
  AffineEquiv.TopologicalSemidirectProduct E

def LeanRidgelet.affineTopologicalMackeySection.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {xi : E}
  (q :
    LeanRidgelet.AffineTopologicalMackeyQuotient
      xi) :
  AffineEquiv.TopologicalSemidirectProduct
    E

Implementation after :=

:=
  SemidirectProduct.inr (Quotient.out q).right

A normalized set-theoretic section of the affine homogeneous quotient.  Starting from
`Quotient.out`, it discards the translation coordinate.  This does not change the coset because
the Mackey subgroup contains every translation, and it is the normalization for which the
section-cocycle character agrees exactly with the Fourier phase.

theoremdefined in LeanRidgelet/HA/AffineMackeyInduction.lean

complete

theorem LeanRidgelet.affineTopologicalMackeySection_rightInverse.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {xi : E} :
  Function.RightInverse LeanRidgelet.affineTopologicalMackeySection
    QuotientGroup.mk

theorem LeanRidgelet.affineTopologicalMackeySection_rightInverse.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {xi : E} :
  Function.RightInverse
    LeanRidgelet.affineTopologicalMackeySection
    QuotientGroup.mk

The normalized affine section is a right inverse of the quotient map.

defdefined in LeanRidgelet/HA/AffineMackeyInduction.lean

complete

def LeanRidgelet.affineTopologicalMackeySectionCocycle.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  {xi : E} (g : AffineEquiv.TopologicalSemidirectProduct E)
  (q : LeanRidgelet.AffineTopologicalMackeyQuotient xi) :
  ↥(LeanRidgelet.affineTopologicalMackeySubgroup xi)

def LeanRidgelet.affineTopologicalMackeySectionCocycle.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {xi : E}
  (g :
    AffineEquiv.TopologicalSemidirectProduct
      E)
  (q :
    LeanRidgelet.AffineTopologicalMackeyQuotient
      xi) :
  ↥(LeanRidgelet.affineTopologicalMackeySubgroup
      xi)

Implementation after :=

:=
  QuotientGroup.leftCosetSectionCocycleOf
    (affineTopologicalMackeySubgroup xi).toSubgroup
    (affineTopologicalMackeySection (E := E))
    affineTopologicalMackeySection_rightInverse g q

The Mackey-subgroup-valued reentry cocycle of the normalized affine section.

defdefined in LeanRidgelet/HA/AffineMackeyInduction.lean

complete

def LeanRidgelet.affineTopologicalMackeySectionPhase.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  {xi : E} (g : AffineEquiv.TopologicalSemidirectProduct E)
  (q : LeanRidgelet.AffineTopologicalMackeyQuotient xi) : ℂ

def LeanRidgelet.affineTopologicalMackeySectionPhase.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {xi : E}
  (g :
    AffineEquiv.TopologicalSemidirectProduct
      E)
  (q :
    LeanRidgelet.AffineTopologicalMackeyQuotient
      xi) :
  ℂ

Implementation after :=

:=
  (affineTopologicalMackeyCharacter xi
    (affineTopologicalMackeySectionCocycle g q) : Circle)

Applying the inducing character to the normalized section cocycle gives the section-model
phase.

theoremdefined in LeanRidgelet/HA/AffineMackeyInduction.lean

complete

theorem LeanRidgelet.affineTopologicalMackeyQuotientHomeomorphDualOrbit_eq_out.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {xi : E} (hxi : xi ≠ 0)
  (q : LeanRidgelet.AffineTopologicalMackeyQuotient xi) :
  ↑((LeanRidgelet.affineTopologicalMackeyQuotientHomeomorphDualOrbit
          hxi)
        q) =
    LeanRidgelet.affineTopologicalDualAction (Quotient.out q).right xi

theorem LeanRidgelet.affineTopologicalMackeyQuotientHomeomorphDualOrbit_eq_out.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {xi : E}
  (hxi : xi ≠ 0)
  (q :
    LeanRidgelet.AffineTopologicalMackeyQuotient
      xi) :
  ↑((LeanRidgelet.affineTopologicalMackeyQuotientHomeomorphDualOrbit
          hxi)
        q) =
    LeanRidgelet.affineTopologicalDualAction
      (Quotient.out q).right xi

Evaluating the quotient-orbit homeomorphism through `Quotient.out` uses only the linear
coordinate of the representative.

theoremdefined in LeanRidgelet/HA/AffineMackeyInduction.lean

complete

theorem LeanRidgelet.affineTopologicalMackeySectionPhase_eq_quotientPhase.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {xi : E} (hxi : xi ≠ 0)
  (g : AffineEquiv.TopologicalSemidirectProduct E)
  (q : LeanRidgelet.AffineTopologicalMackeyQuotient xi) :
  LeanRidgelet.affineTopologicalMackeySectionPhase g q =
    LeanRidgelet.affineTopologicalMackeyQuotientPhase hxi g q

theorem LeanRidgelet.affineTopologicalMackeySectionPhase_eq_quotientPhase.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {xi : E}
  (hxi : xi ≠ 0)
  (g :
    AffineEquiv.TopologicalSemidirectProduct
      E)
  (q :
    LeanRidgelet.AffineTopologicalMackeyQuotient
      xi) :
  LeanRidgelet.affineTopologicalMackeySectionPhase
      g q =
    LeanRidgelet.affineTopologicalMackeyQuotientPhase
      hxi g q

The phase obtained from the normalized section cocycle is exactly the explicit Fourier phase.
Thus the homogeneous quotient representation already is the equivariant-section realization of
unitary induction from `affineTopologicalMackeyCharacter`; no additional gauge multiplier is
needed.

theoremdefined in LeanRidgelet/HA/AffineMackeyInduction.lean

complete

theorem LeanRidgelet.affineTopologicalMackeySectionPhase_cocycle.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {xi : E} (hxi : xi ≠ 0)
  (g h : AffineEquiv.TopologicalSemidirectProduct E)
  (q : LeanRidgelet.AffineTopologicalMackeyQuotient xi) :
  LeanRidgelet.affineTopologicalMackeySectionPhase (g * h) q =
    LeanRidgelet.affineTopologicalMackeySectionPhase g q *
      LeanRidgelet.affineTopologicalMackeySectionPhase h (g⁻¹ • q)

theorem LeanRidgelet.affineTopologicalMackeySectionPhase_cocycle.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {xi : E}
  (hxi : xi ≠ 0)
  (g h :
    AffineEquiv.TopologicalSemidirectProduct
      E)
  (q :
    LeanRidgelet.AffineTopologicalMackeyQuotient
      xi) :
  LeanRidgelet.affineTopologicalMackeySectionPhase
      (g * h) q =
    LeanRidgelet.affineTopologicalMackeySectionPhase
        g q *
      LeanRidgelet.affineTopologicalMackeySectionPhase
        h (g⁻¹ • q)

The normalized section phase is a multiplier cocycle, as follows either from the general
section-cocycle law or from its identification with the Fourier phase.

theoremdefined in LeanRidgelet/HA/AffineMackeyInduction.lean

complete

theorem LeanRidgelet.affineTopologicalMackeySectionPhase_one.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {xi : E} (hxi : xi ≠ 0)
  (q : LeanRidgelet.AffineTopologicalMackeyQuotient xi) :
  LeanRidgelet.affineTopologicalMackeySectionPhase 1 q = 1

theorem LeanRidgelet.affineTopologicalMackeySectionPhase_one.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {xi : E}
  (hxi : xi ≠ 0)
  (q :
    LeanRidgelet.AffineTopologicalMackeyQuotient
      xi) :
  LeanRidgelet.affineTopologicalMackeySectionPhase
      1 q =
    1

The normalized section phase is one at the identity.

theoremdefined in LeanRidgelet/HA/AffineMackeyInduction.lean

complete

theorem LeanRidgelet.continuous_affineTopologicalMackeySectionPhase.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {xi : E} (hxi : xi ≠ 0)
  (g : AffineEquiv.TopologicalSemidirectProduct E) :
  Continuous (LeanRidgelet.affineTopologicalMackeySectionPhase g)

theorem LeanRidgelet.continuous_affineTopologicalMackeySectionPhase.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {xi : E}
  (hxi : xi ≠ 0)
  (g :
    AffineEquiv.TopologicalSemidirectProduct
      E) :
  Continuous
    (LeanRidgelet.affineTopologicalMackeySectionPhase
      g)

Although the chosen normalized section itself is only set-theoretic, its character phase is
continuous: the explicit equality with the Fourier phase supplies the regularity.

theoremdefined in LeanRidgelet/HA/AffineMackeyInduction.lean

complete

theorem LeanRidgelet.continuous_uncurry_affineTopologicalMackeySectionPhase.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {xi : E} (hxi : xi ≠ 0) :
  Continuous
    (Function.uncurry LeanRidgelet.affineTopologicalMackeySectionPhase)

theorem LeanRidgelet.continuous_uncurry_affineTopologicalMackeySectionPhase.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {xi : E}
  (hxi : xi ≠ 0) :
  Continuous
    (Function.uncurry
      LeanRidgelet.affineTopologicalMackeySectionPhase)

The normalized section phase is jointly continuous.  Rewriting it as the explicit Fourier phase
separates the translation coordinate of the group element from the frequency attached to the coset,
and both depend continuously on the pair.

theoremdefined in LeanRidgelet/HA/AffineMackeyInduction.lean

complete

theorem LeanRidgelet.affineTopologicalMackeySectionPhase_norm_one.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {xi : E} (hxi : xi ≠ 0)
  (g : AffineEquiv.TopologicalSemidirectProduct E)
  (q : LeanRidgelet.AffineTopologicalMackeyQuotient xi) :
  ‖LeanRidgelet.affineTopologicalMackeySectionPhase g q‖ = 1

theorem LeanRidgelet.affineTopologicalMackeySectionPhase_norm_one.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {xi : E}
  (hxi : xi ≠ 0)
  (g :
    AffineEquiv.TopologicalSemidirectProduct
      E)
  (q :
    LeanRidgelet.AffineTopologicalMackeyQuotient
      xi) :
  ‖LeanRidgelet.affineTopologicalMackeySectionPhase
        g q‖ =
    1

The normalized section phase is pointwise unimodular.

defdefined in LeanRidgelet/HA/AffineMackeyInduction.lean

complete

def LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E] [MeasurableSpace E]
  [BorelSpace E] {xi : E} (hxi : xi ≠ 0) :
  UnitaryRepresentation (AffineEquiv.TopologicalSemidirectProduct E)
    ↥(MeasureTheory.Lp ℂ 2
        (LeanRidgelet.affineTopologicalMackeyQuotientMeasure hxi))

def LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E]
  [MeasurableSpace E] [BorelSpace E]
  {xi : E} (hxi : xi ≠ 0) :
  UnitaryRepresentation
    (AffineEquiv.TopologicalSemidirectProduct
      E)
    ↥(MeasureTheory.Lp ℂ 2
        (LeanRidgelet.affineTopologicalMackeyQuotientMeasure
          hxi))

Implementation after :=

:=
  twistedQuasiInvariantLpUnitaryRepresentation
    (affineTopologicalMackeyQuotientJacobian (xi := xi))
    measurable_affineTopologicalMackeyQuotient_smul
    (affineTopologicalMackeyQuotientMeasure_map_eq_withDensity hxi)
    affineTopologicalMackeyQuotientJacobian_measurable
    affineTopologicalMackeyQuotientJacobian_ne_zero
    affineTopologicalMackeyQuotientJacobian_one
    affineTopologicalMackeyQuotientJacobian_cocycle
    (affineTopologicalMackeySectionPhase (xi := xi))
    (fun g ↦ (continuous_affineTopologicalMackeySectionPhase hxi g).aestronglyMeasurable)
    (fun g ↦ Filter.Eventually.of_forall
      (affineTopologicalMackeySectionPhase_norm_one hxi g))
    (affineTopologicalMackeySectionPhase_one hxi)
    (affineTopologicalMackeySectionPhase_cocycle hxi)

The normalized-section realization of the unitary representation induced from the Mackey
character.  The reentry factor is literally the inducing character applied to
`s(q)⁻¹ g s(g⁻¹q)`; the quasi-invariant quotient measure supplies the independent
Radon--Nikodym square-root correction.

theoremdefined in LeanRidgelet/HA/AffineMackeyInduction.lean

complete

theorem LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation_apply_ae_explicit.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E] [MeasurableSpace E]
  [BorelSpace E] {xi : E} (hxi : xi ≠ 0)
  (g : AffineEquiv.TopologicalSemidirectProduct E)
  (f :
    ↥(MeasureTheory.Lp ℂ 2
        (LeanRidgelet.affineTopologicalMackeyQuotientMeasure hxi))) :
  ↑↑(↑((LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation
                hxi)
              g)
          f) =ᵐ[LeanRidgelet.affineTopologicalMackeyQuotientMeasure hxi]
    fun q ↦
    LeanRidgelet.affineTopologicalMackeySectionPhase g q *
      (↑↑(NNReal.sqrt (LeanRidgelet.affineTopologicalJacobian g)) *
        ↑↑f (g⁻¹ • q))

theorem LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation_apply_ae_explicit.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E]
  [MeasurableSpace E] [BorelSpace E]
  {xi : E} (hxi : xi ≠ 0)
  (g :
    AffineEquiv.TopologicalSemidirectProduct
      E)
  (f :
    ↥(MeasureTheory.Lp ℂ 2
        (LeanRidgelet.affineTopologicalMackeyQuotientMeasure
          hxi))) :
  ↑↑(↑((LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation
                hxi)
              g)
          f) =ᵐ[LeanRidgelet.affineTopologicalMackeyQuotientMeasure
      hxi]
    fun q ↦
    LeanRidgelet.affineTopologicalMackeySectionPhase
        g q *
      (↑↑(NNReal.sqrt
              (LeanRidgelet.affineTopologicalJacobian
                g)) *
        ↑↑f (g⁻¹ • q))

The normalized-section induced representation has the same explicit action, with the
character of the reentry cocycle displayed through its equality with the Fourier phase.

theoremdefined in LeanRidgelet/HA/AffineMackeyInduction.lean

complete

theorem LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation_eq_quotient.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E] [MeasurableSpace E]
  [BorelSpace E] {xi : E} (hxi : xi ≠ 0) :
  LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation
      hxi =
    LeanRidgelet.affineTopologicalMackeyQuotientCharacterTwistedLpUnitaryRepresentation
      hxi

theorem LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation_eq_quotient.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E]
  [MeasurableSpace E] [BorelSpace E]
  {xi : E} (hxi : xi ≠ 0) :
  LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation
      hxi =
    LeanRidgelet.affineTopologicalMackeyQuotientCharacterTwistedLpUnitaryRepresentation
      hxi

The normalized-section induced model equals the explicit character-twisted quotient
representation.  This closes the model-identification part of unitary induction; the remaining
Mackey endpoint is the irreducibility/imprimitivity theorem itself.

theoremdefined in LeanRidgelet/HA/AffineMackey.lean

complete

theorem LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation_eq_transported.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E] [MeasurableSpace E]
  [BorelSpace E] {xi : E} (hxi : xi ≠ 0) :
  LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation
      hxi =
    LeanRidgelet.affineTopologicalMackeyQuotientLpUnitaryRepresentation
      hxi

theorem LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation_eq_transported.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E]
  [MeasurableSpace E] [BorelSpace E]
  {xi : E} (hxi : xi ≠ 0) :
  LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation
      hxi =
    LeanRidgelet.affineTopologicalMackeyQuotientLpUnitaryRepresentation
      hxi

The normalized-section induced model is exactly the independently transported Fourier
representation on the homogeneous quotient.

theoremdefined in LeanRidgelet/HA/AffineMackey.lean

complete

theorem LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation_isStronglyContinuous.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E] [MeasurableSpace E]
  [BorelSpace E] {xi : E} (hxi : xi ≠ 0) :
  (LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation
      hxi).IsStronglyContinuous

theorem LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation_isStronglyContinuous.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E]
  [MeasurableSpace E] [BorelSpace E]
  {xi : E} (hxi : xi ≠ 0) :
  (LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation
      hxi).IsStronglyContinuous

The normalized-section induced representation is strongly continuous.

defdefined in LeanRidgelet/HA/AffineMackey.lean

complete

def LeanRidgelet.affineMackeySmoothedVector.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [Nontrivial E] [MeasurableSpace E] [BorelSpace E] {xi : E}
  (hxi : xi ≠ 0)
  (ψ :
    CompactlySupportedContinuousMap
      (AffineEquiv.TopologicalSemidirectProduct E) ℂ)
  (f :
    ↥(MeasureTheory.Lp ℂ 2
        (LeanRidgelet.affineTopologicalMackeyQuotientMeasure hxi))) :
  ↥(MeasureTheory.Lp ℂ 2
      (LeanRidgelet.affineTopologicalMackeyQuotientMeasure hxi))

def LeanRidgelet.affineMackeySmoothedVector.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E]
  [MeasurableSpace E] [BorelSpace E]
  {xi : E} (hxi : xi ≠ 0)
  (ψ :
    CompactlySupportedContinuousMap
      (AffineEquiv.TopologicalSemidirectProduct
        E)
      ℂ)
  (f :
    ↥(MeasureTheory.Lp ℂ 2
        (LeanRidgelet.affineTopologicalMackeyQuotientMeasure
          hxi))) :
  ↥(MeasureTheory.Lp ℂ 2
      (LeanRidgelet.affineTopologicalMackeyQuotientMeasure
        hxi))

Implementation after :=

:=
  (affineTopologicalMackeySectionInducedLpUnitaryRepresentation hxi).haarIntegratedVector ψ f

Folland's compactly supported convolution smoothing, expressed intrinsically as a Haar-
integrated vector of the normalized-section induced representation.

theoremdefined in LeanRidgelet/HA/AffineMackey.lean

complete

theorem LeanRidgelet.affineMackey_smoothing_integrable.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [Nontrivial E] [MeasurableSpace E] [BorelSpace E] {xi : E}
  (hxi : xi ≠ 0)
  (ψ :
    CompactlySupportedContinuousMap
      (AffineEquiv.TopologicalSemidirectProduct E) ℂ)
  (f :
    ↥(MeasureTheory.Lp ℂ 2
        (LeanRidgelet.affineTopologicalMackeyQuotientMeasure hxi))) :
  MeasureTheory.Integrable
    (fun g ↦
      ψ g •
        ↑((LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation
                hxi)
              g)
          f)
    MeasureTheory.Measure.haar

theorem LeanRidgelet.affineMackey_smoothing_integrable.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E]
  [MeasurableSpace E] [BorelSpace E]
  {xi : E} (hxi : xi ≠ 0)
  (ψ :
    CompactlySupportedContinuousMap
      (AffineEquiv.TopologicalSemidirectProduct
        E)
      ℂ)
  (f :
    ↥(MeasureTheory.Lp ℂ 2
        (LeanRidgelet.affineTopologicalMackeyQuotientMeasure
          hxi))) :
  MeasureTheory.Integrable
    (fun g ↦
      ψ g •
        ↑((LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation
                hxi)
              g)
          f)
    MeasureTheory.Measure.haar

The orbit-kernel defining affine Mackey smoothing is Bochner integrable.

theoremdefined in LeanRidgelet/HA/AffineMackey.lean

complete

theorem LeanRidgelet.affineMackeySmoothedVector_mem.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [Nontrivial E] [MeasurableSpace E] [BorelSpace E] {xi : E}
  (hxi : xi ≠ 0)
  (K :
    ClosedSubmodule ℂ
      ↥(MeasureTheory.Lp ℂ 2
          (LeanRidgelet.affineTopologicalMackeyQuotientMeasure hxi)))
  (hrepresentation :
    (LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation
          hxi).Commutes
      (↑K).starProjection)
  (ψ :
    CompactlySupportedContinuousMap
      (AffineEquiv.TopologicalSemidirectProduct E) ℂ)
  {f :
    ↥(MeasureTheory.Lp ℂ 2
        (LeanRidgelet.affineTopologicalMackeyQuotientMeasure hxi))}
  (hf : f ∈ K) : LeanRidgelet.affineMackeySmoothedVector hxi ψ f ∈ K

theorem LeanRidgelet.affineMackeySmoothedVector_mem.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E]
  [MeasurableSpace E] [BorelSpace E]
  {xi : E} (hxi : xi ≠ 0)
  (K :
    ClosedSubmodule ℂ
      ↥(MeasureTheory.Lp ℂ 2
          (LeanRidgelet.affineTopologicalMackeyQuotientMeasure
            hxi)))
  (hrepresentation :
    (LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation
          hxi).Commutes
      (↑K).starProjection)
  (ψ :
    CompactlySupportedContinuousMap
      (AffineEquiv.TopologicalSemidirectProduct
        E)
      ℂ)
  {f :
    ↥(MeasureTheory.Lp ℂ 2
        (LeanRidgelet.affineTopologicalMackeyQuotientMeasure
          hxi))}
  (hf : f ∈ K) :
  LeanRidgelet.affineMackeySmoothedVector
      hxi ψ f ∈
    K

Haar convolution smoothing preserves every closed subspace invariant under the induced
representation.

theoremdefined in LeanRidgelet/HA/AffineMackey.lean

complete

theorem LeanRidgelet.affineMackey_exists_smoothing_mem_tendsto.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E] [MeasurableSpace E]
  [BorelSpace E] {xi : E} (hxi : xi ≠ 0)
  (K :
    ClosedSubmodule ℂ
      ↥(MeasureTheory.Lp ℂ 2
          (LeanRidgelet.affineTopologicalMackeyQuotientMeasure hxi)))
  (hrepresentation :
    (LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation
          hxi).Commutes
      (↑K).starProjection)
  {f :
    ↥(MeasureTheory.Lp ℂ 2
        (LeanRidgelet.affineTopologicalMackeyQuotientMeasure hxi))}
  (hf : f ∈ K) :
  ∃ ψ,
    (∀ (n : ℕ) (g : AffineEquiv.TopologicalSemidirectProduct E),
        0 ≤ (ψ n) g) ∧
      (∀ (n : ℕ),
          ∫ (g : AffineEquiv.TopologicalSemidirectProduct E),
              (ψ n) g ∂MeasureTheory.Measure.haar =
            1) ∧
        (∀ U ∈ nhds 1,
            ∀ᶠ (n : ℕ) in Filter.atTop, tsupport ⇑(ψ n) ⊆ U) ∧
          (∀ (n : ℕ),
              LeanRidgelet.affineMackeySmoothedVector hxi
                  (UnitaryRepresentation.complexKernel (ψ n)) f ∈
                K) ∧
            Filter.Tendsto
              (fun n ↦
                LeanRidgelet.affineMackeySmoothedVector hxi
                  (UnitaryRepresentation.complexKernel (ψ n)) f)
              Filter.atTop (nhds f)

theorem LeanRidgelet.affineMackey_exists_smoothing_mem_tendsto.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E]
  [MeasurableSpace E] [BorelSpace E]
  {xi : E} (hxi : xi ≠ 0)
  (K :
    ClosedSubmodule ℂ
      ↥(MeasureTheory.Lp ℂ 2
          (LeanRidgelet.affineTopologicalMackeyQuotientMeasure
            hxi)))
  (hrepresentation :
    (LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation
          hxi).Commutes
      (↑K).starProjection)
  {f :
    ↥(MeasureTheory.Lp ℂ 2
        (LeanRidgelet.affineTopologicalMackeyQuotientMeasure
          hxi))}
  (hf : f ∈ K) :
  ∃ ψ,
    (∀ (n : ℕ)
        (g :
          AffineEquiv.TopologicalSemidirectProduct
            E),
        0 ≤ (ψ n) g) ∧
      (∀ (n : ℕ),
          ∫ (g :
              AffineEquiv.TopologicalSemidirectProduct
                E),
              (ψ n)
                g ∂MeasureTheory.Measure.haar =
            1) ∧
        (∀ U ∈ nhds 1,
            ∀ᶠ (n : ℕ) in Filter.atTop,
              tsupport ⇑(ψ n) ⊆ U) ∧
          (∀ (n : ℕ),
              LeanRidgelet.affineMackeySmoothedVector
                  hxi
                  (UnitaryRepresentation.complexKernel
                    (ψ n))
                  f ∈
                K) ∧
            Filter.Tendsto
              (fun n ↦
                LeanRidgelet.affineMackeySmoothedVector
                  hxi
                  (UnitaryRepresentation.complexKernel
                    (ψ n))
                  f)
              Filter.atTop (nhds f)

Every vector in an invariant closed subspace is the `L²` limit of Haar-smoothed vectors in
the same subspace, using compactly supported nonnegative probability kernels whose supports shrink
to the identity.  This is the approximate-identity part of Folland Lemma 6.29.

theoremdefined in LeanRidgelet/HA/AffineMackey.lean

complete

theorem LeanRidgelet.affineMackeySmoothedVector_exists_continuousRepresentative.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E] [MeasurableSpace E]
  [BorelSpace E] {xi : E} (hxi : xi ≠ 0)
  (ψ :
    CompactlySupportedContinuousMap
      (AffineEquiv.TopologicalSemidirectProduct E) ℂ)
  (f :
    ↥(MeasureTheory.Lp ℂ 2
        (LeanRidgelet.affineTopologicalMackeyQuotientMeasure hxi))) :
  ∃ g,
    Continuous g ∧
      MeasureTheory.MemLp g 2
          (LeanRidgelet.affineTopologicalMackeyQuotientMeasure hxi) ∧
        ↑↑(LeanRidgelet.affineMackeySmoothedVector hxi ψ
                f) =ᵐ[LeanRidgelet.affineTopologicalMackeyQuotientMeasure
            hxi]
          g

theorem LeanRidgelet.affineMackeySmoothedVector_exists_continuousRepresentative.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E]
  [MeasurableSpace E] [BorelSpace E]
  {xi : E} (hxi : xi ≠ 0)
  (ψ :
    CompactlySupportedContinuousMap
      (AffineEquiv.TopologicalSemidirectProduct
        E)
      ℂ)
  (f :
    ↥(MeasureTheory.Lp ℂ 2
        (LeanRidgelet.affineTopologicalMackeyQuotientMeasure
          hxi))) :
  ∃ g,
    Continuous g ∧
      MeasureTheory.MemLp g 2
          (LeanRidgelet.affineTopologicalMackeyQuotientMeasure
            hxi) ∧
        ↑↑(LeanRidgelet.affineMackeySmoothedVector
                hxi ψ
                f) =ᵐ[LeanRidgelet.affineTopologicalMackeyQuotientMeasure
            hxi]
          g

Haar smoothing of the normalized-section induced model has a continuous representative on the
homogeneous quotient.  A measurable representative of the quotient class lifts to the group so that
the smoothing integral becomes a compact-kernel group convolution: the convolution is continuous by
`LeanRidgelet.continuous_affineMackeySmoothingIntegral`, its slices over the finite-measure parts
of the quotient are integrable by
`LeanRidgelet.integrable_uncurry_affineMackeySmoothingIntegrand`, and
`MeasureTheory.integral_L2_coeFn_ae_of_restrict` therefore identifies the `L²`-valued Bochner
integral defining the smoothed vector with that pointwise integral.  The subsequent compact cutoff
and density argument is proved below.

theoremdefined in LeanRidgelet/HA/AffineMackey.lean

complete

theorem LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation_isTopologicallyIrreducible_iff.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E] [MeasurableSpace E]
  [BorelSpace E] {xi : E} (hxi : xi ≠ 0) :
  (LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation
        hxi).IsTopologicallyIrreducible ↔
    (LeanRidgelet.affineDataLpUnitaryRepresentation
        MeasureTheory.volume).IsTopologicallyIrreducible

theorem LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation_isTopologicallyIrreducible_iff.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E]
  [MeasurableSpace E] [BorelSpace E]
  {xi : E} (hxi : xi ≠ 0) :
  (LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation
        hxi).IsTopologicallyIrreducible ↔
    (LeanRidgelet.affineDataLpUnitaryRepresentation
        MeasureTheory.volume).IsTopologicallyIrreducible

Mackey irreducibility for the normalized-section induced model is exactly the remaining
irreducibility statement for the physical affine representation.

theoremdefined in LeanRidgelet/HA/AffineMackeyInduction.lean

complete

theorem LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation_indicator_covariant.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E] [MeasurableSpace E]
  [BorelSpace E] {xi : E} (hxi : xi ≠ 0)
  (g : AffineEquiv.TopologicalSemidirectProduct E)
  (s : Set (LeanRidgelet.AffineTopologicalMackeyQuotient xi))
  (hs : MeasurableSet s) :
  let t := (fun q ↦ g • q) ⁻¹' s;
  MeasureTheory.indicatorLp s hs ∘SL
      ↑((LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation
            hxi)
          g) =
    ↑((LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation
            hxi)
          g) ∘SL
      MeasureTheory.indicatorLp t ⋯

theorem LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation_indicator_covariant.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E]
  [MeasurableSpace E] [BorelSpace E]
  {xi : E} (hxi : xi ≠ 0)
  (g :
    AffineEquiv.TopologicalSemidirectProduct
      E)
  (s :
    Set
      (LeanRidgelet.AffineTopologicalMackeyQuotient
        xi))
  (hs : MeasurableSet s) :
  let t := (fun q ↦ g • q) ⁻¹' s;
  MeasureTheory.indicatorLp s hs ∘SL
      ↑((LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation
            hxi)
          g) =
    ↑((LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation
            hxi)
          g) ∘SL
      MeasureTheory.indicatorLp t ⋯

The measurable-set multiplication projections on the homogeneous quotient form the canonical
system of imprimitivity for the normalized-section induced representation.  In operator form,
restricting the output to `s` and then acting by `g` is the same as first restricting to the
inverse translate of `s` and then acting by `g`.

theoremdefined in LeanRidgelet/HA/AffineMackey.lean

complete

theorem LeanRidgelet.affineTopologicalTranslation_right.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  (b : E) :
  ((AffineEquiv.topologicalSemidirectProductEquiv E).symm
        (LeanRidgelet.affineTranslation b)).right =
    1

theorem LeanRidgelet.affineTopologicalTranslation_right.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (b : E) :
  ((AffineEquiv.topologicalSemidirectProductEquiv
            E).symm
        (LeanRidgelet.affineTranslation
          b)).right =
    1

In topological semidirect-product coordinates, a pure translation has identity linear
coordinate.

theoremdefined in LeanRidgelet/HA/AffineMackey.lean

complete

theorem LeanRidgelet.affineTopologicalMackeyQuotient_translation_smul.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {xi : E} (hxi : xi ≠ 0) (b : E)
  (q : LeanRidgelet.AffineTopologicalMackeyQuotient xi) :
  (AffineEquiv.topologicalSemidirectProductEquiv E).symm
        (LeanRidgelet.affineTranslation b) •
      q =
    q

theorem LeanRidgelet.affineTopologicalMackeyQuotient_translation_smul.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {xi : E}
  (hxi : xi ≠ 0) (b : E)
  (q :
    LeanRidgelet.AffineTopologicalMackeyQuotient
      xi) :
  (AffineEquiv.topologicalSemidirectProductEquiv
            E).symm
        (LeanRidgelet.affineTranslation
          b) •
      q =
    q

The translation subgroup acts trivially on the affine homogeneous quotient.  This is the
quotient-side form of the fact that the orbit map retains only the linear coordinate of an affine
transformation.

theoremdefined in LeanRidgelet/HA/AffineMackey.lean

complete

theorem LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation_translation_apply_ae.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E] [MeasurableSpace E]
  [BorelSpace E] {xi : E} (hxi : xi ≠ 0) (b : E)
  (f :
    ↥(MeasureTheory.Lp ℂ 2
        (LeanRidgelet.affineTopologicalMackeyQuotientMeasure hxi))) :
  ↑↑(↑((LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation
                hxi)
              ((AffineEquiv.topologicalSemidirectProductEquiv E).symm
                (LeanRidgelet.affineTranslation b)))
          f) =ᵐ[LeanRidgelet.affineTopologicalMackeyQuotientMeasure hxi]
    fun q ↦
    ↑((LeanRidgelet.affineTranslationCharacter
            ↑((LeanRidgelet.affineTopologicalMackeyQuotientHomeomorphDualOrbit
                  hxi)
                q))
          b) *
      ↑↑f q

theorem LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation_translation_apply_ae.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E]
  [MeasurableSpace E] [BorelSpace E]
  {xi : E} (hxi : xi ≠ 0) (b : E)
  (f :
    ↥(MeasureTheory.Lp ℂ 2
        (LeanRidgelet.affineTopologicalMackeyQuotientMeasure
          hxi))) :
  ↑↑(↑((LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation
                hxi)
              ((AffineEquiv.topologicalSemidirectProductEquiv
                    E).symm
                (LeanRidgelet.affineTranslation
                  b)))
          f) =ᵐ[LeanRidgelet.affineTopologicalMackeyQuotientMeasure
      hxi]
    fun q ↦
    ↑((LeanRidgelet.affineTranslationCharacter
            ↑((LeanRidgelet.affineTopologicalMackeyQuotientHomeomorphDualOrbit
                  hxi)
                q))
          b) *
      ↑↑f q

The restriction of the normalized-section induced representation to translations is the
pointwise translation-character representation on the frequency orbit.  In particular, both the
homogeneous-space motion and the Radon--Nikodym factor disappear; only the character at the
frequency represented by the coset remains.

defdefined in LeanRidgelet/HA/AffineMackey.lean

complete

def LeanRidgelet.affineTopologicalMackeyFrequencyEmbedding.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {xi : E} (hxi : xi ≠ 0) :
  LeanRidgelet.AffineTopologicalMackeyQuotient xi → E

def LeanRidgelet.affineTopologicalMackeyFrequencyEmbedding.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {xi : E}
  (hxi : xi ≠ 0) :
  LeanRidgelet.AffineTopologicalMackeyQuotient
      xi →
    E

Implementation after :=

:=
  fun q ↦ (affineTopologicalMackeyQuotientHomeomorphDualOrbit hxi q).1

The quotient-orbit coordinate, regarded as a measurable embedding into the ambient frequency
space.

theoremdefined in LeanRidgelet/HA/AffineMackey.lean

complete

theorem LeanRidgelet.affineTopologicalMackeyFrequencyEmbedding_measurableEmbedding.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] {xi : E}
  (hxi : xi ≠ 0) :
  MeasurableEmbedding
    (LeanRidgelet.affineTopologicalMackeyFrequencyEmbedding hxi)

theorem LeanRidgelet.affineTopologicalMackeyFrequencyEmbedding_measurableEmbedding.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  {xi : E} (hxi : xi ≠ 0) :
  MeasurableEmbedding
    (LeanRidgelet.affineTopologicalMackeyFrequencyEmbedding
      hxi)

The quotient-orbit coordinate is a measurable embedding.

defdefined in LeanRidgelet/HA/AffineMackey.lean

complete

def LeanRidgelet.affineTopologicalMackeyTranslationMultiplier.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] {xi : E}
  (hxi : xi ≠ 0) (b : E) :
  ↥(MeasureTheory.Lp ℂ 2
        (LeanRidgelet.affineTopologicalMackeyQuotientMeasure
          hxi)) ≃ₗᵢ[ℂ]
    ↥(MeasureTheory.Lp ℂ 2
        (LeanRidgelet.affineTopologicalMackeyQuotientMeasure hxi))

def LeanRidgelet.affineTopologicalMackeyTranslationMultiplier.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  {xi : E} (hxi : xi ≠ 0) (b : E) :
  ↥(MeasureTheory.Lp ℂ 2
        (LeanRidgelet.affineTopologicalMackeyQuotientMeasure
          hxi)) ≃ₗᵢ[ℂ]
    ↥(MeasureTheory.Lp ℂ 2
        (LeanRidgelet.affineTopologicalMackeyQuotientMeasure
          hxi))

Implementation after :=

:=
  MeasureTheory.fourierCharacterLpMultiplier
    (μ := affineTopologicalMackeyQuotientMeasure hxi)
    (affineTopologicalMackeyFrequencyEmbedding hxi)
    (affineTopologicalMackeyFrequencyEmbedding_measurableEmbedding hxi).measurable b

Multiplication by the character through which a pure translation acts on the Mackey
quotient.  Bundling this operator separately makes the spectral-projection step independent of
the group-representation implementation: its only input is commutation with these concrete
Fourier-character multipliers.

theoremdefined in LeanRidgelet/HA/AffineMackey.lean

complete

theorem LeanRidgelet.affineTopologicalMackeyTranslationMultiplier_apply_ae.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] {xi : E}
  (hxi : xi ≠ 0) (b : E)
  (f :
    ↥(MeasureTheory.Lp ℂ 2
        (LeanRidgelet.affineTopologicalMackeyQuotientMeasure hxi))) :
  ↑↑((LeanRidgelet.affineTopologicalMackeyTranslationMultiplier hxi b)
          f) =ᵐ[LeanRidgelet.affineTopologicalMackeyQuotientMeasure hxi]
    fun q ↦
    ↑((LeanRidgelet.affineTranslationCharacter
            ↑((LeanRidgelet.affineTopologicalMackeyQuotientHomeomorphDualOrbit
                  hxi)
                q))
          b) *
      ↑↑f q

theorem LeanRidgelet.affineTopologicalMackeyTranslationMultiplier_apply_ae.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  {xi : E} (hxi : xi ≠ 0) (b : E)
  (f :
    ↥(MeasureTheory.Lp ℂ 2
        (LeanRidgelet.affineTopologicalMackeyQuotientMeasure
          hxi))) :
  ↑↑((LeanRidgelet.affineTopologicalMackeyTranslationMultiplier
            hxi b)
          f) =ᵐ[LeanRidgelet.affineTopologicalMackeyQuotientMeasure
      hxi]
    fun q ↦
    ↑((LeanRidgelet.affineTranslationCharacter
            ↑((LeanRidgelet.affineTopologicalMackeyQuotientHomeomorphDualOrbit
                  hxi)
                q))
          b) *
      ↑↑f q

The bundled Mackey translation multiplier has the expected pointwise representative.

theoremdefined in LeanRidgelet/HA/AffineMackey.lean

complete

theorem LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation_translation_eq_multiplier.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E] [MeasurableSpace E]
  [BorelSpace E] {xi : E} (hxi : xi ≠ 0) (b : E) :
  ↑((LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation
          hxi)
        ((AffineEquiv.topologicalSemidirectProductEquiv E).symm
          (LeanRidgelet.affineTranslation b))) =
    ↑↑(LeanRidgelet.affineTopologicalMackeyTranslationMultiplier hxi b)

theorem LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation_translation_eq_multiplier.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E]
  [MeasurableSpace E] [BorelSpace E]
  {xi : E} (hxi : xi ≠ 0) (b : E) :
  ↑((LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation
          hxi)
        ((AffineEquiv.topologicalSemidirectProductEquiv
              E).symm
          (LeanRidgelet.affineTranslation
            b))) =
    ↑↑(LeanRidgelet.affineTopologicalMackeyTranslationMultiplier
          hxi b)

On the translation subgroup, the normalized-section induced representation is the concrete
Mackey character multiplier.

theoremdefined in LeanRidgelet/HA/AffineMackey.lean

complete

theorem LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation_commutes_translation.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E] [MeasurableSpace E]
  [BorelSpace E] {xi : E} (hxi : xi ≠ 0)
  (T :
    ↥(MeasureTheory.Lp ℂ 2
          (LeanRidgelet.affineTopologicalMackeyQuotientMeasure
            hxi)) →L[ℂ]
      ↥(MeasureTheory.Lp ℂ 2
          (LeanRidgelet.affineTopologicalMackeyQuotientMeasure hxi)))
  (hT :
    (LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation
          hxi).Commutes
      T)
  (b : E) :
  T ∘SL
      ↑((LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation
            hxi)
          ((AffineEquiv.topologicalSemidirectProductEquiv E).symm
            (LeanRidgelet.affineTranslation b))) =
    ↑((LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation
            hxi)
          ((AffineEquiv.topologicalSemidirectProductEquiv E).symm
            (LeanRidgelet.affineTranslation b))) ∘SL
      T

theorem LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation_commutes_translation.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E]
  [MeasurableSpace E] [BorelSpace E]
  {xi : E} (hxi : xi ≠ 0)
  (T :
    ↥(MeasureTheory.Lp ℂ 2
          (LeanRidgelet.affineTopologicalMackeyQuotientMeasure
            hxi)) →L[ℂ]
      ↥(MeasureTheory.Lp ℂ 2
          (LeanRidgelet.affineTopologicalMackeyQuotientMeasure
            hxi)))
  (hT :
    (LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation
          hxi).Commutes
      T)
  (b : E) :
  T ∘SL
      ↑((LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation
            hxi)
          ((AffineEquiv.topologicalSemidirectProductEquiv
                E).symm
            (LeanRidgelet.affineTranslation
              b))) =
    ↑((LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation
            hxi)
          ((AffineEquiv.topologicalSemidirectProductEquiv
                E).symm
            (LeanRidgelet.affineTranslation
              b))) ∘SL
      T

An operator in the commutant of the induced affine representation commutes, in particular,
with every member of its translation restriction.  Together with the preceding pointwise formula,
this is the operator-theoretic input to the spectral-projection step.

theoremdefined in LeanRidgelet/HA/AffineMackey.lean

complete

theorem LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation_commutes_translationMultiplier.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E] [MeasurableSpace E]
  [BorelSpace E] {xi : E} (hxi : xi ≠ 0)
  (T :
    ↥(MeasureTheory.Lp ℂ 2
          (LeanRidgelet.affineTopologicalMackeyQuotientMeasure
            hxi)) →L[ℂ]
      ↥(MeasureTheory.Lp ℂ 2
          (LeanRidgelet.affineTopologicalMackeyQuotientMeasure hxi)))
  (hT :
    (LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation
          hxi).Commutes
      T)
  (b : E) :
  T ∘SL
      ↑↑(LeanRidgelet.affineTopologicalMackeyTranslationMultiplier hxi
            b) =
    ↑↑(LeanRidgelet.affineTopologicalMackeyTranslationMultiplier hxi
            b) ∘SL
      T

theorem LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation_commutes_translationMultiplier.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E]
  [MeasurableSpace E] [BorelSpace E]
  {xi : E} (hxi : xi ≠ 0)
  (T :
    ↥(MeasureTheory.Lp ℂ 2
          (LeanRidgelet.affineTopologicalMackeyQuotientMeasure
            hxi)) →L[ℂ]
      ↥(MeasureTheory.Lp ℂ 2
          (LeanRidgelet.affineTopologicalMackeyQuotientMeasure
            hxi)))
  (hT :
    (LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation
          hxi).Commutes
      T)
  (b : E) :
  T ∘SL
      ↑↑(LeanRidgelet.affineTopologicalMackeyTranslationMultiplier
            hxi b) =
    ↑↑(LeanRidgelet.affineTopologicalMackeyTranslationMultiplier
            hxi b) ∘SL
      T

An operator in the affine commutant therefore commutes with every concrete translation
character multiplier.

theoremdefined in LeanRidgelet/HA/AffineMackey.lean

complete

theorem LeanRidgelet.affineMackey_commutes_indicator_of_commutes_translation.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] {xi : E}
  (hxi : xi ≠ 0)
  (T :
    ↥(MeasureTheory.Lp ℂ 2
          (LeanRidgelet.affineTopologicalMackeyQuotientMeasure
            hxi)) →L[ℂ]
      ↥(MeasureTheory.Lp ℂ 2
          (LeanRidgelet.affineTopologicalMackeyQuotientMeasure hxi)))
  (htranslation :
    ∀ (b : E),
      T ∘SL
          ↑↑(LeanRidgelet.affineTopologicalMackeyTranslationMultiplier
                hxi b) =
        ↑↑(LeanRidgelet.affineTopologicalMackeyTranslationMultiplier hxi
                b) ∘SL
          T)
  (s : Set (LeanRidgelet.AffineTopologicalMackeyQuotient xi))
  (hs : MeasurableSet s) :
  T ∘SL MeasureTheory.indicatorLp s hs =
    MeasureTheory.indicatorLp s hs ∘SL T

theorem LeanRidgelet.affineMackey_commutes_indicator_of_commutes_translation.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  {xi : E} (hxi : xi ≠ 0)
  (T :
    ↥(MeasureTheory.Lp ℂ 2
          (LeanRidgelet.affineTopologicalMackeyQuotientMeasure
            hxi)) →L[ℂ]
      ↥(MeasureTheory.Lp ℂ 2
          (LeanRidgelet.affineTopologicalMackeyQuotientMeasure
            hxi)))
  (htranslation :
    ∀ (b : E),
      T ∘SL
          ↑↑(LeanRidgelet.affineTopologicalMackeyTranslationMultiplier
                hxi b) =
        ↑↑(LeanRidgelet.affineTopologicalMackeyTranslationMultiplier
                hxi b) ∘SL
          T)
  (s :
    Set
      (LeanRidgelet.AffineTopologicalMackeyQuotient
        xi))
  (hs : MeasurableSet s) :
  T ∘SL MeasureTheory.indicatorLp s hs =
    MeasureTheory.indicatorLp s hs ∘SL T

Specialization of the commutant criterion in Folland Theorem 4.44 to the translation
restriction of the affine induced model.  Since translations act by the characters displayed in
`affineTopologicalMackeySectionInducedLpUnitaryRepresentation_translation_apply_ae`, their spectral
projections are the canonical measurable-set multiplication operators on the quotient orbit.

theoremdefined in LeanRidgelet/HA/AffineMackey.lean

complete

theorem LeanRidgelet.affineMackey_indicatorLp_mem.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E] {xi : E} (hxi : xi ≠ 0)
  (K :
    ClosedSubmodule ℂ
      ↥(MeasureTheory.Lp ℂ 2
          (LeanRidgelet.affineTopologicalMackeyQuotientMeasure hxi)))
  (hindicator :
    ∀ (s : Set (LeanRidgelet.AffineTopologicalMackeyQuotient xi))
      (hs : MeasurableSet s),
      (↑K).starProjection ∘SL MeasureTheory.indicatorLp s hs =
        MeasureTheory.indicatorLp s hs ∘SL (↑K).starProjection)
  (s : Set (LeanRidgelet.AffineTopologicalMackeyQuotient xi))
  (hs : MeasurableSet s)
  {f :
    ↥(MeasureTheory.Lp ℂ 2
        (LeanRidgelet.affineTopologicalMackeyQuotientMeasure hxi))}
  (hf : f ∈ K) : (MeasureTheory.indicatorLp s hs) f ∈ K

theorem LeanRidgelet.affineMackey_indicatorLp_mem.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  {xi : E} (hxi : xi ≠ 0)
  (K :
    ClosedSubmodule ℂ
      ↥(MeasureTheory.Lp ℂ 2
          (LeanRidgelet.affineTopologicalMackeyQuotientMeasure
            hxi)))
  (hindicator :
    ∀
      (s :
        Set
          (LeanRidgelet.AffineTopologicalMackeyQuotient
            xi))
      (hs : MeasurableSet s),
      (↑K).starProjection ∘SL
          MeasureTheory.indicatorLp s hs =
        MeasureTheory.indicatorLp s hs ∘SL
          (↑K).starProjection)
  (s :
    Set
      (LeanRidgelet.AffineTopologicalMackeyQuotient
        xi))
  (hs : MeasurableSet s)
  {f :
    ↥(MeasureTheory.Lp ℂ 2
        (LeanRidgelet.affineTopologicalMackeyQuotientMeasure
          hxi))}
  (hf : f ∈ K) :
  (MeasureTheory.indicatorLp s hs) f ∈ K

A system-invariant closed subspace is stable under every quotient-orbit indicator
projection.  This is the elementary range-of-projection consequence used before the regular
section argument of Folland Lemma 6.29.

theoremdefined in LeanRidgelet/HA/AffineMackey.lean

complete

theorem LeanRidgelet.affineMackey_representation_mem.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [Nontrivial E] [MeasurableSpace E] [BorelSpace E] {xi : E}
  (hxi : xi ≠ 0)
  (K :
    ClosedSubmodule ℂ
      ↥(MeasureTheory.Lp ℂ 2
          (LeanRidgelet.affineTopologicalMackeyQuotientMeasure hxi)))
  (hrepresentation :
    (LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation
          hxi).Commutes
      (↑K).starProjection)
  (g : AffineEquiv.TopologicalSemidirectProduct E)
  {f :
    ↥(MeasureTheory.Lp ℂ 2
        (LeanRidgelet.affineTopologicalMackeyQuotientMeasure hxi))}
  (hf : f ∈ K) :
  ↑((LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation
            hxi)
          g)
      f ∈
    K

theorem LeanRidgelet.affineMackey_representation_mem.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E]
  [MeasurableSpace E] [BorelSpace E]
  {xi : E} (hxi : xi ≠ 0)
  (K :
    ClosedSubmodule ℂ
      ↥(MeasureTheory.Lp ℂ 2
          (LeanRidgelet.affineTopologicalMackeyQuotientMeasure
            hxi)))
  (hrepresentation :
    (LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation
          hxi).Commutes
      (↑K).starProjection)
  (g :
    AffineEquiv.TopologicalSemidirectProduct
      E)
  {f :
    ↥(MeasureTheory.Lp ℂ 2
        (LeanRidgelet.affineTopologicalMackeyQuotientMeasure
          hxi))}
  (hf : f ∈ K) :
  ↑((LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation
            hxi)
          g)
      f ∈
    K

A closed subspace whose star projection commutes with the induced representation is stable
under the induced action.

theoremdefined in LeanRidgelet/HA/AffineMackey.lean

complete

theorem LeanRidgelet.affineMackey_regularSection_memLp.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E] {xi : E} (hxi : xi ≠ 0)
  (f :
    CompactlySupportedContinuousMap
      (LeanRidgelet.AffineTopologicalMackeyQuotient xi) ℂ) :
  MeasureTheory.MemLp (⇑f) 2
    (LeanRidgelet.affineTopologicalMackeyQuotientMeasure hxi)

theorem LeanRidgelet.affineMackey_regularSection_memLp.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  {xi : E} (hxi : xi ≠ 0)
  (f :
    CompactlySupportedContinuousMap
      (LeanRidgelet.AffineTopologicalMackeyQuotient
        xi)
      ℂ) :
  MeasureTheory.MemLp (⇑f) 2
    (LeanRidgelet.affineTopologicalMackeyQuotientMeasure
      hxi)

A compactly supported continuous scalar section on the homogeneous quotient belongs to
`L²`.

defdefined in LeanRidgelet/HA/AffineMackey.lean

complete

def LeanRidgelet.affineMackeyRegularSectionToLp.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E] {xi : E} (hxi : xi ≠ 0) :
  CompactlySupportedContinuousMap
      (LeanRidgelet.AffineTopologicalMackeyQuotient xi) ℂ →ₗ[ℂ]
    ↥(MeasureTheory.Lp ℂ 2
        (LeanRidgelet.affineTopologicalMackeyQuotientMeasure hxi))

def LeanRidgelet.affineMackeyRegularSectionToLp.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  {xi : E} (hxi : xi ≠ 0) :
  CompactlySupportedContinuousMap
      (LeanRidgelet.AffineTopologicalMackeyQuotient
        xi)
      ℂ →ₗ[ℂ]
    ↥(MeasureTheory.Lp ℂ 2
        (LeanRidgelet.affineTopologicalMackeyQuotientMeasure
          hxi))

Implementation after :=

:=
  compactlySupportedContinuousToLp

The linear map sending a regular quotient section to its `L²` class.

defdefined in LeanRidgelet/HA/AffineMackey.lean

complete

def LeanRidgelet.affineMackeyRegularSectionEvaluation.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  {xi : E} :
  CompactlySupportedContinuousMap
      (LeanRidgelet.AffineTopologicalMackeyQuotient xi) ℂ →ₗ[ℂ]
    ℂ

def LeanRidgelet.affineMackeyRegularSectionEvaluation.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {xi : E} :
  CompactlySupportedContinuousMap
      (LeanRidgelet.AffineTopologicalMackeyQuotient
        xi)
      ℂ →ₗ[ℂ]
    ℂ

Implementation after :=

:= f (QuotientGroup.mk (1 : AffineEquiv.TopologicalSemidirectProduct E))
  map_add' f g := by simp
  map_smul' c f := by simp

Evaluation of a regular section at the identity coset.

defdefined in LeanRidgelet/HA/AffineMackey.lean

complete

def LeanRidgelet.affineMackeyRegularSectionsIn.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E] {xi : E} (hxi : xi ≠ 0)
  (K :
    ClosedSubmodule ℂ
      ↥(MeasureTheory.Lp ℂ 2
          (LeanRidgelet.affineTopologicalMackeyQuotientMeasure hxi))) :
  Submodule ℂ
    (CompactlySupportedContinuousMap
      (LeanRidgelet.AffineTopologicalMackeyQuotient xi) ℂ)

def LeanRidgelet.affineMackeyRegularSectionsIn.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  {xi : E} (hxi : xi ≠ 0)
  (K :
    ClosedSubmodule ℂ
      ↥(MeasureTheory.Lp ℂ 2
          (LeanRidgelet.affineTopologicalMackeyQuotientMeasure
            hxi))) :
  Submodule ℂ
    (CompactlySupportedContinuousMap
      (LeanRidgelet.AffineTopologicalMackeyQuotient
        xi)
      ℂ)

Implementation after :=

:=
  K.toSubmodule.comap (affineMackeyRegularSectionToLp hxi)

Regular sections whose `L²` classes belong to a fixed closed system-invariant subspace.

defdefined in LeanRidgelet/HA/AffineMackey.lean

complete

def LeanRidgelet.affineMackeyInducingFiber.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E] {xi : E} (hxi : xi ≠ 0)
  (K :
    ClosedSubmodule ℂ
      ↥(MeasureTheory.Lp ℂ 2
          (LeanRidgelet.affineTopologicalMackeyQuotientMeasure hxi))) :
  ClosedSubmodule ℂ ℂ

def LeanRidgelet.affineMackeyInducingFiber.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  {xi : E} (hxi : xi ≠ 0)
  (K :
    ClosedSubmodule ℂ
      ↥(MeasureTheory.Lp ℂ 2
          (LeanRidgelet.affineTopologicalMackeyQuotientMeasure
            hxi))) :
  ClosedSubmodule ℂ ℂ

Implementation after :=

:=
  (Submodule.map affineMackeyRegularSectionEvaluation
    (affineMackeyRegularSectionsIn hxi K)).closure

The inducing fiber extracted from a closed subspace: take the closed linear span of the
values at the identity coset of its regular sections.  This is the concrete scalar specialization
of the fiber `M` constructed in Folland Lemma 6.30.

theoremdefined in LeanRidgelet/HA/AffineMackey.lean

complete

theorem LeanRidgelet.affineMackey_regularSection_dense.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [Nontrivial E] [MeasurableSpace E] [BorelSpace E] {xi : E}
  (hxi : xi ≠ 0)
  (K :
    ClosedSubmodule ℂ
      ↥(MeasureTheory.Lp ℂ 2
          (LeanRidgelet.affineTopologicalMackeyQuotientMeasure hxi)))
  (hrepresentation :
    (LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation
          hxi).Commutes
      (↑K).starProjection)
  (hindicator :
    ∀ (s : Set (LeanRidgelet.AffineTopologicalMackeyQuotient xi))
      (hs : MeasurableSet s),
      (↑K).starProjection ∘SL MeasureTheory.indicatorLp s hs =
        MeasureTheory.indicatorLp s hs ∘SL (↑K).starProjection) :
  Dense
    (Set.range fun f ↦
      ⟨(LeanRidgelet.affineMackeyRegularSectionToLp hxi) ↑f, ⋯⟩)

theorem LeanRidgelet.affineMackey_regularSection_dense.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E]
  [MeasurableSpace E] [BorelSpace E]
  {xi : E} (hxi : xi ≠ 0)
  (K :
    ClosedSubmodule ℂ
      ↥(MeasureTheory.Lp ℂ 2
          (LeanRidgelet.affineTopologicalMackeyQuotientMeasure
            hxi)))
  (hrepresentation :
    (LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation
          hxi).Commutes
      (↑K).starProjection)
  (hindicator :
    ∀
      (s :
        Set
          (LeanRidgelet.AffineTopologicalMackeyQuotient
            xi))
      (hs : MeasurableSet s),
      (↑K).starProjection ∘SL
          MeasureTheory.indicatorLp s hs =
        MeasureTheory.indicatorLp s hs ∘SL
          (↑K).starProjection) :
  Dense
    (Set.range fun f ↦
      ⟨(LeanRidgelet.affineMackeyRegularSectionToLp
            hxi)
          ↑f,
        ⋯⟩)

Folland Lemma 6.29 in the normalized quotient model: regular sections belonging to a closed
subspace invariant under the system of imprimitivity are dense in that subspace.

defdefined in LeanRidgelet/HA/AffineMackey.lean

complete

def LeanRidgelet.affineMackeyRegularSectionSMul.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  {xi : E} (hxi : xi ≠ 0)
  (g : AffineEquiv.TopologicalSemidirectProduct E)
  (r :
    CompactlySupportedContinuousMap
      (LeanRidgelet.AffineTopologicalMackeyQuotient xi) ℂ) :
  CompactlySupportedContinuousMap
    (LeanRidgelet.AffineTopologicalMackeyQuotient xi) ℂ

def LeanRidgelet.affineMackeyRegularSectionSMul.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {xi : E}
  (hxi : xi ≠ 0)
  (g :
    AffineEquiv.TopologicalSemidirectProduct
      E)
  (r :
    CompactlySupportedContinuousMap
      (LeanRidgelet.AffineTopologicalMackeyQuotient
        xi)
      ℂ) :
  CompactlySupportedContinuousMap
    (LeanRidgelet.AffineTopologicalMackeyQuotient
      xi)
    ℂ

Implementation after :=

:= affineTopologicalMackeySectionPhase g q *
    (((affineTopologicalJacobian g).sqrt : ℂ) * r (g⁻¹ • q))
  continuous_toFun :=
    (continuous_affineTopologicalMackeySectionPhase hxi g).mul
      (continuous_const.mul (r.continuous.comp (continuous_const_smul g⁻¹)))
  hasCompactSupport' := by
    have h : HasCompactSupport
        (fun q : AffineTopologicalMackeyQuotient xi ↦ r (g⁻¹ • q)) :=
      r.hasCompactSupport.comp_homeomorph (Homeomorph.smul g⁻¹)
    have heq : (fun q : AffineTopologicalMackeyQuotient xi ↦
          affineTopologicalMackeySectionPhase g q *
            (((affineTopologicalJacobian g).sqrt : ℂ) * r (g⁻¹ • q))) =
        (fun q : AffineTopologicalMackeyQuotient xi ↦
          affineTopologicalMackeySectionPhase g q *
            ((affineTopologicalJacobian g).sqrt : ℂ)) *
          fun q : AffineTopologicalMackeyQuotient xi ↦ r (g⁻¹ • q) := by
      funext q
      simp only [Pi.mul_apply, mul_assoc]
    rw [heq]
    exact h.mul_left

Translating a compactly supported continuous quotient section by the induced action.  The
phase and the Radon--Nikodym constant are unit-modulus and positive, so the translate is again a
compactly supported continuous section.

theoremdefined in LeanRidgelet/HA/AffineMackey.lean

complete

theorem LeanRidgelet.affineMackeyRegularSectionSMul_apply.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  {xi : E} (hxi : xi ≠ 0)
  (g : AffineEquiv.TopologicalSemidirectProduct E)
  (r :
    CompactlySupportedContinuousMap
      (LeanRidgelet.AffineTopologicalMackeyQuotient xi) ℂ)
  (q : LeanRidgelet.AffineTopologicalMackeyQuotient xi) :
  (LeanRidgelet.affineMackeyRegularSectionSMul hxi g r) q =
    LeanRidgelet.affineTopologicalMackeySectionPhase g q *
      (↑↑(NNReal.sqrt (LeanRidgelet.affineTopologicalJacobian g)) *
        r (g⁻¹ • q))

theorem LeanRidgelet.affineMackeyRegularSectionSMul_apply.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {xi : E}
  (hxi : xi ≠ 0)
  (g :
    AffineEquiv.TopologicalSemidirectProduct
      E)
  (r :
    CompactlySupportedContinuousMap
      (LeanRidgelet.AffineTopologicalMackeyQuotient
        xi)
      ℂ)
  (q :
    LeanRidgelet.AffineTopologicalMackeyQuotient
      xi) :
  (LeanRidgelet.affineMackeyRegularSectionSMul
        hxi g r)
      q =
    LeanRidgelet.affineTopologicalMackeySectionPhase
        g q *
      (↑↑(NNReal.sqrt
              (LeanRidgelet.affineTopologicalJacobian
                g)) *
        r (g⁻¹ • q))

theoremdefined in LeanRidgelet/HA/AffineMackey.lean

complete

theorem LeanRidgelet.affineMackeyRegularSectionToLp_coeFn_ae.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] {xi : E}
  (hxi : xi ≠ 0)
  (r :
    CompactlySupportedContinuousMap
      (LeanRidgelet.AffineTopologicalMackeyQuotient xi) ℂ) :
  ↑↑((LeanRidgelet.affineMackeyRegularSectionToLp hxi)
          r) =ᵐ[LeanRidgelet.affineTopologicalMackeyQuotientMeasure hxi]
    ⇑r

theorem LeanRidgelet.affineMackeyRegularSectionToLp_coeFn_ae.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  {xi : E} (hxi : xi ≠ 0)
  (r :
    CompactlySupportedContinuousMap
      (LeanRidgelet.AffineTopologicalMackeyQuotient
        xi)
      ℂ) :
  ↑↑((LeanRidgelet.affineMackeyRegularSectionToLp
            hxi)
          r) =ᵐ[LeanRidgelet.affineTopologicalMackeyQuotientMeasure
      hxi]
    ⇑r

A compactly supported continuous section represents its own `L²` class.

theoremdefined in LeanRidgelet/HA/AffineMackey.lean

complete

theorem LeanRidgelet.affineMackeyRegularSectionToLp_smul.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [Nontrivial E] [MeasurableSpace E] [BorelSpace E] {xi : E}
  (hxi : xi ≠ 0) (g : AffineEquiv.TopologicalSemidirectProduct E)
  (r :
    CompactlySupportedContinuousMap
      (LeanRidgelet.AffineTopologicalMackeyQuotient xi) ℂ) :
  (LeanRidgelet.affineMackeyRegularSectionToLp hxi)
      (LeanRidgelet.affineMackeyRegularSectionSMul hxi g r) =
    ↑((LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation
            hxi)
          g)
      ((LeanRidgelet.affineMackeyRegularSectionToLp hxi) r)

theorem LeanRidgelet.affineMackeyRegularSectionToLp_smul.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E]
  [MeasurableSpace E] [BorelSpace E]
  {xi : E} (hxi : xi ≠ 0)
  (g :
    AffineEquiv.TopologicalSemidirectProduct
      E)
  (r :
    CompactlySupportedContinuousMap
      (LeanRidgelet.AffineTopologicalMackeyQuotient
        xi)
      ℂ) :
  (LeanRidgelet.affineMackeyRegularSectionToLp
        hxi)
      (LeanRidgelet.affineMackeyRegularSectionSMul
        hxi g r) =
    ↑((LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation
            hxi)
          g)
      ((LeanRidgelet.affineMackeyRegularSectionToLp
          hxi)
        r)

The `L²` class of a translated section is the induced action applied to the `L²` class of the
section.

theoremdefined in LeanRidgelet/HA/AffineMackey.lean

complete

theorem LeanRidgelet.affineMackeyRegularSectionSMul_ne_zero.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  {xi : E} (hxi : xi ≠ 0)
  (g : AffineEquiv.TopologicalSemidirectProduct E)
  (r :
    CompactlySupportedContinuousMap
      (LeanRidgelet.AffineTopologicalMackeyQuotient xi) ℂ)
  {q : LeanRidgelet.AffineTopologicalMackeyQuotient xi}
  (hq : r (g⁻¹ • q) ≠ 0) :
  (LeanRidgelet.affineMackeyRegularSectionSMul hxi g r) q ≠ 0

theorem LeanRidgelet.affineMackeyRegularSectionSMul_ne_zero.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {xi : E}
  (hxi : xi ≠ 0)
  (g :
    AffineEquiv.TopologicalSemidirectProduct
      E)
  (r :
    CompactlySupportedContinuousMap
      (LeanRidgelet.AffineTopologicalMackeyQuotient
        xi)
      ℂ)
  {q :
    LeanRidgelet.AffineTopologicalMackeyQuotient
      xi}
  (hq : r (g⁻¹ • q) ≠ 0) :
  (LeanRidgelet.affineMackeyRegularSectionSMul
        hxi g r)
      q ≠
    0

The phase and Radon--Nikodym factors of the section model never vanish, so a translated section
vanishes exactly where the original section does.

theoremdefined in LeanRidgelet/HA/AffineMackey.lean

complete

theorem LeanRidgelet.affineMackeyRegularSection_eq_zero_of_smul_eq_zero.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {xi : E} (hxi : xi ≠ 0)
  (g : AffineEquiv.TopologicalSemidirectProduct E)
  (r :
    CompactlySupportedContinuousMap
      (LeanRidgelet.AffineTopologicalMackeyQuotient xi) ℂ)
  {q : LeanRidgelet.AffineTopologicalMackeyQuotient xi}
  (h : (LeanRidgelet.affineMackeyRegularSectionSMul hxi g r) q = 0) :
  r (g⁻¹ • q) = 0

theorem LeanRidgelet.affineMackeyRegularSection_eq_zero_of_smul_eq_zero.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {xi : E}
  (hxi : xi ≠ 0)
  (g :
    AffineEquiv.TopologicalSemidirectProduct
      E)
  (r :
    CompactlySupportedContinuousMap
      (LeanRidgelet.AffineTopologicalMackeyQuotient
        xi)
      ℂ)
  {q :
    LeanRidgelet.AffineTopologicalMackeyQuotient
      xi}
  (h :
    (LeanRidgelet.affineMackeyRegularSectionSMul
          hxi g r)
        q =
      0) :
  r (g⁻¹ • q) = 0

Vanishing of a translated section at the identity coset means vanishing of the section at the
translated coset.

theoremdefined in LeanRidgelet/HA/AffineMackey.lean

complete

theorem LeanRidgelet.affineMackeyInducingFiber_eq_bot_iff.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E] {xi : E} (hxi : xi ≠ 0)
  (K :
    ClosedSubmodule ℂ
      ↥(MeasureTheory.Lp ℂ 2
          (LeanRidgelet.affineTopologicalMackeyQuotientMeasure hxi))) :
  LeanRidgelet.affineMackeyInducingFiber hxi K = ⊥ ↔
    ∀
      (r :
        CompactlySupportedContinuousMap
          (LeanRidgelet.AffineTopologicalMackeyQuotient xi) ℂ),
      (LeanRidgelet.affineMackeyRegularSectionToLp hxi) r ∈ K → r ↑1 = 0

theorem LeanRidgelet.affineMackeyInducingFiber_eq_bot_iff.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  {xi : E} (hxi : xi ≠ 0)
  (K :
    ClosedSubmodule ℂ
      ↥(MeasureTheory.Lp ℂ 2
          (LeanRidgelet.affineTopologicalMackeyQuotientMeasure
            hxi))) :
  LeanRidgelet.affineMackeyInducingFiber
        hxi K =
      ⊥ ↔
    ∀
      (r :
        CompactlySupportedContinuousMap
          (LeanRidgelet.AffineTopologicalMackeyQuotient
            xi)
          ℂ),
      (LeanRidgelet.affineMackeyRegularSectionToLp
              hxi)
            r ∈
          K →
        r ↑1 = 0

The inducing fiber is trivial exactly when every regular section of the subspace vanishes at
the identity coset.

theoremdefined in LeanRidgelet/HA/AffineMackey.lean

complete

theorem LeanRidgelet.affineMackeyInducingFiber_eq_top_of_ne_zero.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] {xi : E}
  (hxi : xi ≠ 0)
  (K :
    ClosedSubmodule ℂ
      ↥(MeasureTheory.Lp ℂ 2
          (LeanRidgelet.affineTopologicalMackeyQuotientMeasure hxi)))
  (r :
    CompactlySupportedContinuousMap
      (LeanRidgelet.AffineTopologicalMackeyQuotient xi) ℂ)
  (hr : (LeanRidgelet.affineMackeyRegularSectionToLp hxi) r ∈ K)
  (hr0 : r ↑1 ≠ 0) : LeanRidgelet.affineMackeyInducingFiber hxi K = ⊤

theorem LeanRidgelet.affineMackeyInducingFiber_eq_top_of_ne_zero.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  {xi : E} (hxi : xi ≠ 0)
  (K :
    ClosedSubmodule ℂ
      ↥(MeasureTheory.Lp ℂ 2
          (LeanRidgelet.affineTopologicalMackeyQuotientMeasure
            hxi)))
  (r :
    CompactlySupportedContinuousMap
      (LeanRidgelet.AffineTopologicalMackeyQuotient
        xi)
      ℂ)
  (hr :
    (LeanRidgelet.affineMackeyRegularSectionToLp
          hxi)
        r ∈
      K)
  (hr0 : r ↑1 ≠ 0) :
  LeanRidgelet.affineMackeyInducingFiber
      hxi K =
    ⊤

One regular section of the subspace that does not vanish at the identity coset already spans
the whole one-dimensional inducing fiber.

theoremdefined in LeanRidgelet/HA/AffineMackey.lean

complete

theorem LeanRidgelet.affineMackey_regularSection_eq_zero_of_inducingFiber_eq_bot.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E] [MeasurableSpace E]
  [BorelSpace E] {xi : E} (hxi : xi ≠ 0)
  (K :
    ClosedSubmodule ℂ
      ↥(MeasureTheory.Lp ℂ 2
          (LeanRidgelet.affineTopologicalMackeyQuotientMeasure hxi)))
  (hrepresentation :
    (LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation
          hxi).Commutes
      (↑K).starProjection)
  (hfiber : LeanRidgelet.affineMackeyInducingFiber hxi K = ⊥)
  (r :
    CompactlySupportedContinuousMap
      (LeanRidgelet.AffineTopologicalMackeyQuotient xi) ℂ)
  (hr : (LeanRidgelet.affineMackeyRegularSectionToLp hxi) r ∈ K)
  (q : LeanRidgelet.AffineTopologicalMackeyQuotient xi) : r q = 0

theorem LeanRidgelet.affineMackey_regularSection_eq_zero_of_inducingFiber_eq_bot.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E]
  [MeasurableSpace E] [BorelSpace E]
  {xi : E} (hxi : xi ≠ 0)
  (K :
    ClosedSubmodule ℂ
      ↥(MeasureTheory.Lp ℂ 2
          (LeanRidgelet.affineTopologicalMackeyQuotientMeasure
            hxi)))
  (hrepresentation :
    (LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation
          hxi).Commutes
      (↑K).starProjection)
  (hfiber :
    LeanRidgelet.affineMackeyInducingFiber
        hxi K =
      ⊥)
  (r :
    CompactlySupportedContinuousMap
      (LeanRidgelet.AffineTopologicalMackeyQuotient
        xi)
      ℂ)
  (hr :
    (LeanRidgelet.affineMackeyRegularSectionToLp
          hxi)
        r ∈
      K)
  (q :
    LeanRidgelet.AffineTopologicalMackeyQuotient
      xi) :
  r q = 0

If the inducing fiber is trivial, then every regular section of the subspace vanishes
identically: the induced action moves the identity coset onto every coset, so the fiber condition
propagates along the orbit.

theoremdefined in LeanRidgelet/HA/AffineMackey.lean

complete

theorem LeanRidgelet.affineMackey_eq_bot_of_inducingFiber_eq_bot.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E] [MeasurableSpace E]
  [BorelSpace E] {xi : E} (hxi : xi ≠ 0)
  (K :
    ClosedSubmodule ℂ
      ↥(MeasureTheory.Lp ℂ 2
          (LeanRidgelet.affineTopologicalMackeyQuotientMeasure hxi)))
  (hrepresentation :
    (LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation
          hxi).Commutes
      (↑K).starProjection)
  (hregular :
    Dense
      (Set.range fun f ↦
        ⟨(LeanRidgelet.affineMackeyRegularSectionToLp hxi) ↑f, ⋯⟩))
  (hfiber : LeanRidgelet.affineMackeyInducingFiber hxi K = ⊥) : K = ⊥

theorem LeanRidgelet.affineMackey_eq_bot_of_inducingFiber_eq_bot.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E]
  [MeasurableSpace E] [BorelSpace E]
  {xi : E} (hxi : xi ≠ 0)
  (K :
    ClosedSubmodule ℂ
      ↥(MeasureTheory.Lp ℂ 2
          (LeanRidgelet.affineTopologicalMackeyQuotientMeasure
            hxi)))
  (hrepresentation :
    (LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation
          hxi).Commutes
      (↑K).starProjection)
  (hregular :
    Dense
      (Set.range fun f ↦
        ⟨(LeanRidgelet.affineMackeyRegularSectionToLp
              hxi)
            ↑f,
          ⋯⟩))
  (hfiber :
    LeanRidgelet.affineMackeyInducingFiber
        hxi K =
      ⊥) :
  K = ⊥

A subspace with trivial inducing fiber is trivial.  Every regular section it contains vanishes
identically, and regular sections are dense in it by the Folland-6.29 density theorem.

theoremdefined in LeanRidgelet/HA/AffineMackey.lean

complete

theorem LeanRidgelet.affineMackey_eq_top_of_regularSection_ne_zero.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E] [MeasurableSpace E]
  [BorelSpace E] {xi : E} (hxi : xi ≠ 0)
  (K :
    ClosedSubmodule ℂ
      ↥(MeasureTheory.Lp ℂ 2
          (LeanRidgelet.affineTopologicalMackeyQuotientMeasure hxi)))
  (hrepresentation :
    (LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation
          hxi).Commutes
      (↑K).starProjection)
  (hindicator :
    ∀ (s : Set (LeanRidgelet.AffineTopologicalMackeyQuotient xi))
      (hs : MeasurableSet s),
      (↑K).starProjection ∘SL MeasureTheory.indicatorLp s hs =
        MeasureTheory.indicatorLp s hs ∘SL (↑K).starProjection)
  (r :
    CompactlySupportedContinuousMap
      (LeanRidgelet.AffineTopologicalMackeyQuotient xi) ℂ)
  (hr : (LeanRidgelet.affineMackeyRegularSectionToLp hxi) r ∈ K)
  (hr0 : r ↑1 ≠ 0) : K = ⊤

theorem LeanRidgelet.affineMackey_eq_top_of_regularSection_ne_zero.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E]
  [MeasurableSpace E] [BorelSpace E]
  {xi : E} (hxi : xi ≠ 0)
  (K :
    ClosedSubmodule ℂ
      ↥(MeasureTheory.Lp ℂ 2
          (LeanRidgelet.affineTopologicalMackeyQuotientMeasure
            hxi)))
  (hrepresentation :
    (LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation
          hxi).Commutes
      (↑K).starProjection)
  (hindicator :
    ∀
      (s :
        Set
          (LeanRidgelet.AffineTopologicalMackeyQuotient
            xi))
      (hs : MeasurableSet s),
      (↑K).starProjection ∘SL
          MeasureTheory.indicatorLp s hs =
        MeasureTheory.indicatorLp s hs ∘SL
          (↑K).starProjection)
  (r :
    CompactlySupportedContinuousMap
      (LeanRidgelet.AffineTopologicalMackeyQuotient
        xi)
      ℂ)
  (hr :
    (LeanRidgelet.affineMackeyRegularSectionToLp
          hxi)
        r ∈
      K)
  (hr0 : r ↑1 ≠ 0) : K = ⊤

A subspace containing one regular section that does not vanish at the identity coset is
everything.  Orthogonality to the subspace forces a vector to vanish almost everywhere on the
nonvanishing set of every translate of that section, and countably many translates already cover the
homogeneous quotient.

theoremdefined in LeanRidgelet/HA/AffineMackey.lean

complete

theorem LeanRidgelet.affineMackey_closedSubspace_extreme_iff_inducingFiber_extreme.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E] [MeasurableSpace E]
  [BorelSpace E] {xi : E} (hxi : xi ≠ 0)
  (K :
    ClosedSubmodule ℂ
      ↥(MeasureTheory.Lp ℂ 2
          (LeanRidgelet.affineTopologicalMackeyQuotientMeasure hxi)))
  (hrepresentation :
    (LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation
          hxi).Commutes
      (↑K).starProjection)
  (hindicator :
    ∀ (s : Set (LeanRidgelet.AffineTopologicalMackeyQuotient xi))
      (hs : MeasurableSet s),
      (↑K).starProjection ∘SL MeasureTheory.indicatorLp s hs =
        MeasureTheory.indicatorLp s hs ∘SL (↑K).starProjection)
  (hregular :
    Dense
      (Set.range fun f ↦
        ⟨(LeanRidgelet.affineMackeyRegularSectionToLp hxi) ↑f, ⋯⟩)) :
  (K = ⊥ ↔ LeanRidgelet.affineMackeyInducingFiber hxi K = ⊥) ∧
    (K = ⊤ ↔ LeanRidgelet.affineMackeyInducingFiber hxi K = ⊤)

theorem LeanRidgelet.affineMackey_closedSubspace_extreme_iff_inducingFiber_extreme.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E]
  [MeasurableSpace E] [BorelSpace E]
  {xi : E} (hxi : xi ≠ 0)
  (K :
    ClosedSubmodule ℂ
      ↥(MeasureTheory.Lp ℂ 2
          (LeanRidgelet.affineTopologicalMackeyQuotientMeasure
            hxi)))
  (hrepresentation :
    (LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation
          hxi).Commutes
      (↑K).starProjection)
  (hindicator :
    ∀
      (s :
        Set
          (LeanRidgelet.AffineTopologicalMackeyQuotient
            xi))
      (hs : MeasurableSet s),
      (↑K).starProjection ∘SL
          MeasureTheory.indicatorLp s hs =
        MeasureTheory.indicatorLp s hs ∘SL
          (↑K).starProjection)
  (hregular :
    Dense
      (Set.range fun f ↦
        ⟨(LeanRidgelet.affineMackeyRegularSectionToLp
              hxi)
            ↑f,
          ⋯⟩)) :
  (K = ⊥ ↔
      LeanRidgelet.affineMackeyInducingFiber
          hxi K =
        ⊥) ∧
    (K = ⊤ ↔
      LeanRidgelet.affineMackeyInducingFiber
          hxi K =
        ⊤)

Folland Lemma 6.30, restricted to the two conclusions needed for the one-dimensional affine
inducing fiber.  The regular-section fiber is zero exactly when the induced closed subspace is
zero, and it is the whole scalar fiber exactly when that subspace is the whole `L²` space.

Both directions are assembled from the four lemmas above: identity-coset evaluation of translated
sections for the zero case, together with the density hypothesis supplied by Folland 6.29, and
orthogonal-complement vanishing along a countable subcover of translates for the full case.

theoremdefined in LeanRidgelet/HA/AffineMackey.lean

complete

theorem LeanRidgelet.affineMackey_systemInvariant_closedSubspace_eq_bot_or_top.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E] [MeasurableSpace E]
  [BorelSpace E] {xi : E} (hxi : xi ≠ 0)
  (K :
    ClosedSubmodule ℂ
      ↥(MeasureTheory.Lp ℂ 2
          (LeanRidgelet.affineTopologicalMackeyQuotientMeasure hxi)))
  (hrepresentation :
    (LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation
          hxi).Commutes
      (↑K).starProjection)
  (hindicator :
    ∀ (s : Set (LeanRidgelet.AffineTopologicalMackeyQuotient xi))
      (hs : MeasurableSet s),
      (↑K).starProjection ∘SL MeasureTheory.indicatorLp s hs =
        MeasureTheory.indicatorLp s hs ∘SL (↑K).starProjection) :
  K = ⊥ ∨ K = ⊤

theorem LeanRidgelet.affineMackey_systemInvariant_closedSubspace_eq_bot_or_top.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E]
  [MeasurableSpace E] [BorelSpace E]
  {xi : E} (hxi : xi ≠ 0)
  (K :
    ClosedSubmodule ℂ
      ↥(MeasureTheory.Lp ℂ 2
          (LeanRidgelet.affineTopologicalMackeyQuotientMeasure
            hxi)))
  (hrepresentation :
    (LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation
          hxi).Commutes
      (↑K).starProjection)
  (hindicator :
    ∀
      (s :
        Set
          (LeanRidgelet.AffineTopologicalMackeyQuotient
            xi))
      (hs : MeasurableSet s),
      (↑K).starProjection ∘SL
          MeasureTheory.indicatorLp s hs =
        MeasureTheory.indicatorLp s hs ∘SL
          (↑K).starProjection) :
  K = ⊥ ∨ K = ⊤

Folland Lemmas 6.29--6.30 for the canonical affine system with one-dimensional inducing
fiber. A closed subspace whose orthogonal projection commutes with both the induced action and all
quotient-orbit indicator projections is zero or the whole `L²` space.

The regular-section density theorem above is a completed consequence of the named
measurable-lift/product-integrability/convolution-formula placeholder, and the inducing-fiber
correspondence is now proved. This theorem combines them with the proved classification of closed
complex subspaces of the one-dimensional inducing fiber and contains no `sorry` of its own.

theoremdefined in LeanRidgelet/HA/AffineMackey.lean

complete

theorem LeanRidgelet.affineMackey_scalar_of_commutes_indicators.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E] [MeasurableSpace E]
  [BorelSpace E] {xi : E} (hxi : xi ≠ 0)
  (T :
    ↥(MeasureTheory.Lp ℂ 2
          (LeanRidgelet.affineTopologicalMackeyQuotientMeasure
            hxi)) →L[ℂ]
      ↥(MeasureTheory.Lp ℂ 2
          (LeanRidgelet.affineTopologicalMackeyQuotientMeasure hxi)))
  (hT :
    (LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation
          hxi).Commutes
      T)
  (hindicator :
    ∀ (s : Set (LeanRidgelet.AffineTopologicalMackeyQuotient xi))
      (hs : MeasurableSet s),
      T ∘SL MeasureTheory.indicatorLp s hs =
        MeasureTheory.indicatorLp s hs ∘SL T) :
  ∃ c,
    T =
      c •
        ContinuousLinearMap.id ℂ
          ↥(MeasureTheory.Lp ℂ 2
              (LeanRidgelet.affineTopologicalMackeyQuotientMeasure hxi))

theorem LeanRidgelet.affineMackey_scalar_of_commutes_indicators.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E]
  [MeasurableSpace E] [BorelSpace E]
  {xi : E} (hxi : xi ≠ 0)
  (T :
    ↥(MeasureTheory.Lp ℂ 2
          (LeanRidgelet.affineTopologicalMackeyQuotientMeasure
            hxi)) →L[ℂ]
      ↥(MeasureTheory.Lp ℂ 2
          (LeanRidgelet.affineTopologicalMackeyQuotientMeasure
            hxi)))
  (hT :
    (LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation
          hxi).Commutes
      T)
  (hindicator :
    ∀
      (s :
        Set
          (LeanRidgelet.AffineTopologicalMackeyQuotient
            xi))
      (hs : MeasurableSet s),
      T ∘SL
          MeasureTheory.indicatorLp s hs =
        MeasureTheory.indicatorLp s hs ∘SL
          T) :
  ∃ c,
    T =
      c •
        ContinuousLinearMap.id ℂ
          ↥(MeasureTheory.Lp ℂ 2
              (LeanRidgelet.affineTopologicalMackeyQuotientMeasure
                hxi))

Specialization of Folland Theorem 6.28 to the canonical affine system of imprimitivity.
Commutation with the induced action and all quotient-orbit indicator projections identifies `T`
with an operator in the commutant of the one-dimensional inducing character, hence with a scalar.

The only unproved input is the Folland-6.29 smoothed-vector continuous-representative theorem
used by the completed regular-section density argument. The spectral-subspace argument, the
inducing-fiber correspondence, and the decomposition into two self-adjoint operators are proved
here.

theoremdefined in LeanRidgelet/HA/AffineMackey.lean

complete

theorem LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation_hasSchurProperty.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E] [MeasurableSpace E]
  [BorelSpace E] {xi : E} (hxi : xi ≠ 0) :
  (LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation
      hxi).HasSchurProperty

theorem LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation_hasSchurProperty.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E]
  [MeasurableSpace E] [BorelSpace E]
  {xi : E} (hxi : xi ≠ 0) :
  (LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation
      hxi).HasSchurProperty

The assembled commutant form of the Mackey irreducibility step for the normalized-section
induced model. An operator commuting with the induced affine action first commutes with the
spectral projections of the translation subgroup, hence belongs to the commutant of the canonical
system of imprimitivity; the imprimitivity commutant theorem then identifies it with the commutant
of the inducing one-dimensional character, so it is scalar.

All group, orbit, quotient-measure, section-cocycle, continuity, inducing-character,
translation-spectral, and inducing-fiber inputs have already been constructed above. The remaining
analytic input is the named Folland-6.29 smoothed-vector continuous-representative theorem used by
the completed density argument; this assembly contains no source-level placeholder and uses no
induction or imprimitivity assumptions structure.

theoremdefined in LeanRidgelet/HA/AffineMackey.lean

complete

theorem LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation_isTopologicallyIrreducible.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E] [MeasurableSpace E]
  [BorelSpace E] {xi : E} (hxi : xi ≠ 0) :
  (LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation
      hxi).IsTopologicallyIrreducible

theorem LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation_isTopologicallyIrreducible.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E]
  [MeasurableSpace E] [BorelSpace E]
  {xi : E} (hxi : xi ≠ 0) :
  (LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation
      hxi).IsTopologicallyIrreducible

Folland Theorem 6.39, irreducibility direction, for the normalized-section affine induced
model.  Once its commutant is scalar, the converse direction of the unitary Schur lemma turns the
commutant statement into topological irreducibility.

theoremdefined in LeanRidgelet/HA/AffineMackey.lean

complete

theorem LeanRidgelet.affineDataLpUnitaryRepresentation_isTopologicallyIrreducible.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E] [MeasurableSpace E]
  [BorelSpace E] :
  (LeanRidgelet.affineDataLpUnitaryRepresentation
      MeasureTheory.volume).IsTopologicallyIrreducible

theorem LeanRidgelet.affineDataLpUnitaryRepresentation_isTopologicallyIrreducible.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E]
  [MeasurableSpace E] [BorelSpace E] :
  (LeanRidgelet.affineDataLpUnitaryRepresentation
      MeasureTheory.volume).IsTopologicallyIrreducible

Theorem 2.5 of arXiv:2405.13682: the scalar quasi-regular representation of the full affine
group is topologically irreducible.

The Fourier-conjugated representation has now been transported first to the intrinsic `L²` space
on the conull nonzero-frequency subtype and then, through
`affineTopologicalMackeyQuotientHomeomorphDualOrbit`, to the actual homogeneous-space
`L²(G/H)`. Both transports have explicit inverse bounded intertwiners, and the homeomorphism is
proved equivariant for left translation and the dual action. The closed locally compact inducing
subgroup and its strongly continuous irreducible character representation are
`affineTopologicalMackeySubgroup` and `affineTopologicalMackeyUnitaryRepresentation`, with the
required translation and little-group restriction formulas.

The normalized equivariant-section realization of unitary induction is now constructed explicitly:
its reentry cocycle is `s(q)⁻¹ g s(g⁻¹q)`, its character phase is proved equal to the Fourier
phase, and the resulting unitary representation is proved equal to the transported homogeneous
model. The paper endpoint is derived from the commutant form of Folland Theorem 6.39 above. Its
translation-spectral, spectral-subspace, compact-kernel convolution continuity, compact-cutoff,
regular-section density, and 6.30 inducing-fiber inputs are complete, while its Folland-6.29
measurable lift with product integrability and convolution formula is the one remaining HA
placeholder. Mathlib's algebraic
`Representation.ind` is not the quasi-invariant Hilbert-space construction and Mathlib has no
imprimitivity theorem.

Inducing fiber. Translating a compactly supported continuous section by the induced action gives another such section, because the phase is continuous and unimodular, the Radon--Nikodym factor is a positive constant, and the shifted support is the homeomorphic image of a compact set; its L^2 class is the induced action applied to the original class. Since the transported quotient measure is positive on nonempty open sets, a section whose class vanishes vanishes identically, so a trivial subspace has a trivial fiber; conversely, if the fiber is trivial then evaluating translated sections at the identity coset kills a section at every coset, because the group acts transitively on the quotient, and regular-section density then forces the subspace to be trivial. A Urysohn bump provides a section with value one at the identity coset, so the full subspace has full fiber. If some section of the subspace does not vanish at the identity coset, then every vector orthogonal to the subspace vanishes almost everywhere on the nonvanishing set of each translate, by the indicator test of the Mathlib candidate layer; those nonvanishing sets are open and cover the quotient, second countability extracts a countable subcover, and the orthogonal complement is therefore zero. The combination of the two lemmas, the assembled Schur property, Schur's converse, and the model transports contain no further source-level sorry and derive the paper theorem.

Haar measure of the affine group along the orbit map

Theorem4.3.7

Statement uses 3

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used by 1✓L∃∀N

Quotient-integral input for the smoothing estimate. Local integrability of the lift of a quotient L^2 class needs an integral over a compact set of group elements to become an integral over a compact set of frequencies. The estimate proved here does exactly that: the image of a compactly restricted Haar measure of the affine group under the orbit map x\mapsto (x_{\mathrm{right}}^{-1})^\dagger\xi is dominated by a finite multiple of Lebesgue measure of frequency space, restricted to a compact set.

Lean code for Theorem4.3.7●9 declarations

defdefined in LeanRidgelet/HA/AffineGroupHaar.lean

complete

def LeanRidgelet.affineLinearDeterminantCharacter.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] :
  (E →L[ℝ] E)ˣ →* ENNReal

def LeanRidgelet.affineLinearDeterminantCharacter.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E] :
  (E →L[ℝ] E)ˣ →* ENNReal

Implementation after :=

:= ENNReal.ofReal |LinearMap.det ((L : E →L[ℝ] E) : E →ₗ[ℝ] E)|⁻¹
  map_one' := by simp
  map_mul' L M := by
    have hdet : LinearMap.det (((L * M : (E →L[ℝ] E)ˣ) : E →L[ℝ] E) : E →ₗ[ℝ] E) =
        LinearMap.det ((L : E →L[ℝ] E) : E →ₗ[ℝ] E) *
          LinearMap.det ((M : E →L[ℝ] E) : E →ₗ[ℝ] E) := by
      change LinearMap.det (((L : E →L[ℝ] E) ∘L (M : E →L[ℝ] E) : E →L[ℝ] E) : E →ₗ[ℝ] E) = _
      exact LinearMap.det_comp _ _
    rw [hdet, abs_mul, mul_inv, ENNReal.ofReal_mul (inv_nonneg.mpr (abs_nonneg _))]

The reciprocal absolute determinant of the linear factor of the affine group, as a
multiplicative character.  It is the factor by which the linear action rescales Lebesgue measure of
the translation factor, hence the density that turns Haar measure of the linear factor into the
relatively invariant measure of the semidirect-product factorization.

theoremdefined in LeanRidgelet/HA/AffineGroupHaar.lean

complete

theorem LeanRidgelet.affineLinearDeterminantCharacter_apply.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] (L : (E →L[ℝ] E)ˣ) :
  LeanRidgelet.affineLinearDeterminantCharacter L =
    ENNReal.ofReal |LinearMap.det ↑↑L|⁻¹

theorem LeanRidgelet.affineLinearDeterminantCharacter_apply.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  (L : (E →L[ℝ] E)ˣ) :
  LeanRidgelet.affineLinearDeterminantCharacter
      L =
    ENNReal.ofReal |LinearMap.det ↑↑L|⁻¹

theoremdefined in LeanRidgelet/HA/AffineGroupHaar.lean

complete

theorem LeanRidgelet.affineLinearDeterminantCharacter_ne_zero.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (L : (E →L[ℝ] E)ˣ) :
  LeanRidgelet.affineLinearDeterminantCharacter L ≠ 0

theorem LeanRidgelet.affineLinearDeterminantCharacter_ne_zero.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  (L : (E →L[ℝ] E)ˣ) :
  LeanRidgelet.affineLinearDeterminantCharacter
      L ≠
    0

theoremdefined in LeanRidgelet/HA/AffineGroupHaar.lean

complete

theorem LeanRidgelet.continuous_affineLinearDeterminantCharacter_real.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] : Continuous fun L ↦ |LinearMap.det ↑↑L|⁻¹

theorem LeanRidgelet.continuous_affineLinearDeterminantCharacter_real.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] :
  Continuous fun L ↦ |LinearMap.det ↑↑L|⁻¹

theoremdefined in LeanRidgelet/HA/AffineGroupHaar.lean

complete

theorem LeanRidgelet.measurable_affineLinearDeterminantCharacter.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace (E →L[ℝ] E)ˣ]
  [BorelSpace (E →L[ℝ] E)ˣ] :
  Measurable ⇑LeanRidgelet.affineLinearDeterminantCharacter

theorem LeanRidgelet.measurable_affineLinearDeterminantCharacter.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace (E →L[ℝ] E)ˣ]
  [BorelSpace (E →L[ℝ] E)ˣ] :
  Measurable
    ⇑LeanRidgelet.affineLinearDeterminantCharacter

theoremdefined in LeanRidgelet/HA/AffineGroupHaar.lean

complete

theorem LeanRidgelet.exists_affineLinearDeterminantCharacter_le_of_isCompact.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {K : Set (E →L[ℝ] E)ˣ} (hK : IsCompact K) :
  ∃ C,
    C ≠ ⊤ ∧ ∀ L ∈ K, LeanRidgelet.affineLinearDeterminantCharacter L ≤ C

theorem LeanRidgelet.exists_affineLinearDeterminantCharacter_le_of_isCompact.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  {K : Set (E →L[ℝ] E)ˣ}
  (hK : IsCompact K) :
  ∃ C,
    C ≠ ⊤ ∧
      ∀ L ∈ K,
        LeanRidgelet.affineLinearDeterminantCharacter
            L ≤
          C

On a compact set of linear factors the character is bounded.

theoremdefined in LeanRidgelet/HA/AffineGroupHaar.lean

complete

theorem LeanRidgelet.affine_map_orbitMap_haar_restrict_le.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E] {xi : E} (hxi : xi ≠ 0)
  {S : Set (AffineEquiv.TopologicalSemidirectProduct E)}
  (hS : IsCompact S) :
  ∃ C B,
    C ≠ ⊤ ∧
      IsCompact B ∧
        MeasureTheory.Measure.map
            (LeanRidgelet.affineTopologicalMackeyOrbitMap xi)
            (MeasureTheory.Measure.haar.restrict S) ≤
          C • MeasureTheory.volume.restrict B

theorem LeanRidgelet.affine_map_orbitMap_haar_restrict_le.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  {xi : E} (hxi : xi ≠ 0)
  {S :
    Set
      (AffineEquiv.TopologicalSemidirectProduct
        E)}
  (hS : IsCompact S) :
  ∃ C B,
    C ≠ ⊤ ∧
      IsCompact B ∧
        MeasureTheory.Measure.map
            (LeanRidgelet.affineTopologicalMackeyOrbitMap
              xi)
            (MeasureTheory.Measure.haar.restrict
              S) ≤
          C •
            MeasureTheory.volume.restrict
              B

The image of a compactly restricted Haar measure of the affine group under the frequency orbit
map is dominated by a finite multiple of Lebesgue measure of frequency space, restricted to the
compact image.

The translation factor contributes only the finite Haar measure of a compact set, because left
translation acts on the two semidirect-product coordinates separately; the linear factor is handled
by the contragredient-orbit estimate.

theoremdefined in LeanRidgelet/HA/AffineGroupHaar.lean

complete

theorem LeanRidgelet.affineTopologicalMackeyOrbitMap_inv.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  (xi : E) (x : AffineEquiv.TopologicalSemidirectProduct E) :
  LeanRidgelet.affineTopologicalMackeyOrbitMap xi x⁻¹ =
    (ContinuousLinearMap.adjointEvalLinearMap E xi) ↑x.right

theorem LeanRidgelet.affineTopologicalMackeyOrbitMap_inv.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] (xi : E)
  (x :
    AffineEquiv.TopologicalSemidirectProduct
      E) :
  LeanRidgelet.affineTopologicalMackeyOrbitMap
      xi x⁻¹ =
    (ContinuousLinearMap.adjointEvalLinearMap
        E xi)
      ↑x.right

The orbit map evaluated at an inverse is the adjoint orbit map: the linear coordinate of an
inverse is the inverse linear coordinate, and the contragredient of an inverse is the adjoint.

theoremdefined in LeanRidgelet/HA/AffineGroupHaar.lean

complete

theorem LeanRidgelet.affine_map_adjointOrbitMap_haar_restrict_le.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] {xi : E}
  (hxi : xi ≠ 0) {S : Set (AffineEquiv.TopologicalSemidirectProduct E)}
  (hS : IsCompact S) :
  ∃ C B,
    C ≠ ⊤ ∧
      IsCompact B ∧
        MeasureTheory.Measure.map
            (fun x ↦
              LeanRidgelet.affineTopologicalMackeyOrbitMap xi x⁻¹)
            (MeasureTheory.Measure.haar.restrict S) ≤
          C • MeasureTheory.volume.restrict B

theorem LeanRidgelet.affine_map_adjointOrbitMap_haar_restrict_le.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  {xi : E} (hxi : xi ≠ 0)
  {S :
    Set
      (AffineEquiv.TopologicalSemidirectProduct
        E)}
  (hS : IsCompact S) :
  ∃ C B,
    C ≠ ⊤ ∧
      IsCompact B ∧
        MeasureTheory.Measure.map
            (fun x ↦
              LeanRidgelet.affineTopologicalMackeyOrbitMap
                xi x⁻¹)
            (MeasureTheory.Measure.haar.restrict
              S) ≤
          C •
            MeasureTheory.volume.restrict
              B

The same bound for the orbit map composed with inversion, which is the adjoint orbit map
`x ↦ (x.right)† ξ`.  This is the form consumed by the group-convolution continuity theorem, whose
local-integrability hypothesis is stated for the inverse-composed integrand.

Taking adjoints is linear on the operator algebra, so this variant does not use the Haar transport
along the contragredient automorphism.

The same bound is proved for the orbit map composed with inversion, that is for the adjoint orbit map x\mapsto x_{\mathrm{right}}^\dagger\xi; that is the form the group-convolution continuity theorem consumes, since its local-integrability hypothesis is stated for the inverse-composed integrand. The proof instantiates the general semidirect-product factorization at the affine group and composes it with the contragredient-orbit estimate. The translation factor Multiplicative E is the additive group of E written multiplicatively, so its Haar measure is Lebesgue measure and the linear action rescales it by the reciprocal absolute determinant; that determinant is the multiplicative character of the factorization, continuous and nonvanishing, hence bounded on compact sets, which converts the relatively invariant measure of the linear factor back to Haar measure there. Borel structures on the two factors are introduced inside the proof rather than as global instances, since neither factor appears in the statement.

The equivariant lift to the group

Definition4.3.8

Statement uses 3

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Lifting a quotient class to the group. Folland realizes an induced representation on functions over the group that transform by the inducing character, which turns smoothing into an ordinary group convolution. Here the lift multiplies a function on the quotient by the inverse section phase at the point itself, F(x)=P(x)^{-1}f(xH) with P(x)=\mathrm{phase}(x,xH).

Lean code for Definition4.3.8●22 declarations

defdefined in LeanRidgelet/HA/AffineMackeyLift.lean

complete

def LeanRidgelet.affineMackeyLiftPhase.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  {xi : E} (x : AffineEquiv.TopologicalSemidirectProduct E) : ℂ

def LeanRidgelet.affineMackeyLiftPhase.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {xi : E}
  (x :
    AffineEquiv.TopologicalSemidirectProduct
      E) :
  ℂ

Implementation after :=

:=
  affineTopologicalMackeySectionPhase (xi := xi) x (QuotientGroup.mk x)

The section phase of a group element at its own coset.  It is the gauge that makes the lift of a
quotient function transform by the inducing character.

theoremdefined in LeanRidgelet/HA/AffineMackeyLift.lean

complete

theorem LeanRidgelet.affineMackeyLiftPhase_norm_one.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  {xi : E} (hxi : xi ≠ 0)
  (x : AffineEquiv.TopologicalSemidirectProduct E) :
  ‖LeanRidgelet.affineMackeyLiftPhase x‖ = 1

theorem LeanRidgelet.affineMackeyLiftPhase_norm_one.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {xi : E}
  (hxi : xi ≠ 0)
  (x :
    AffineEquiv.TopologicalSemidirectProduct
      E) :
  ‖LeanRidgelet.affineMackeyLiftPhase x‖ =
    1

theoremdefined in LeanRidgelet/HA/AffineMackeyLift.lean

complete

theorem LeanRidgelet.affineMackeyLiftPhase_ne_zero.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  {xi : E} (hxi : xi ≠ 0)
  (x : AffineEquiv.TopologicalSemidirectProduct E) :
  LeanRidgelet.affineMackeyLiftPhase x ≠ 0

theorem LeanRidgelet.affineMackeyLiftPhase_ne_zero.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {xi : E}
  (hxi : xi ≠ 0)
  (x :
    AffineEquiv.TopologicalSemidirectProduct
      E) :
  LeanRidgelet.affineMackeyLiftPhase x ≠ 0

theoremdefined in LeanRidgelet/HA/AffineMackeyLift.lean

complete

theorem LeanRidgelet.continuous_affineMackeyLiftPhase.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  {xi : E} (hxi : xi ≠ 0) :
  Continuous LeanRidgelet.affineMackeyLiftPhase

theorem LeanRidgelet.continuous_affineMackeyLiftPhase.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {xi : E}
  (hxi : xi ≠ 0) :
  Continuous
    LeanRidgelet.affineMackeyLiftPhase

The lift phase is continuous: although the normalized section itself is only set-theoretic, its
character equals the explicit Fourier phase, which is jointly continuous.

theoremdefined in LeanRidgelet/HA/AffineMackeyLift.lean

complete

theorem LeanRidgelet.affineMackeyLiftPhase_mul.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  {xi : E} (hxi : xi ≠ 0)
  (g x : AffineEquiv.TopologicalSemidirectProduct E) :
  LeanRidgelet.affineMackeyLiftPhase x =
    LeanRidgelet.affineTopologicalMackeySectionPhase g ↑x *
      LeanRidgelet.affineMackeyLiftPhase (g⁻¹ * x)

theorem LeanRidgelet.affineMackeyLiftPhase_mul.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {xi : E}
  (hxi : xi ≠ 0)
  (g x :
    AffineEquiv.TopologicalSemidirectProduct
      E) :
  LeanRidgelet.affineMackeyLiftPhase x =
    LeanRidgelet.affineTopologicalMackeySectionPhase
        g ↑x *
      LeanRidgelet.affineMackeyLiftPhase
        (g⁻¹ * x)

The cocycle law of the lift phase: translating the argument by `g⁻¹` splits off exactly the
section phase of `g` at the coset.

defdefined in LeanRidgelet/HA/AffineMackeyLift.lean

complete

def LeanRidgelet.affineMackeyLiftFun.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  {xi : E} (f : LeanRidgelet.AffineTopologicalMackeyQuotient xi → ℂ)
  (x : AffineEquiv.TopologicalSemidirectProduct E) : ℂ

def LeanRidgelet.affineMackeyLiftFun.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {xi : E}
  (f :
    LeanRidgelet.AffineTopologicalMackeyQuotient
        xi →
      ℂ)
  (x :
    AffineEquiv.TopologicalSemidirectProduct
      E) :
  ℂ

Implementation after :=

:=
  (affineMackeyLiftPhase (xi := xi) x)⁻¹ * f (QuotientGroup.mk x)

The lift of a function on the affine homogeneous quotient to the affine group.

theoremdefined in LeanRidgelet/HA/AffineMackeyLift.lean

complete

theorem LeanRidgelet.affineMackeyLiftFun_apply.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  {xi : E} (f : LeanRidgelet.AffineTopologicalMackeyQuotient xi → ℂ)
  (x : AffineEquiv.TopologicalSemidirectProduct E) :
  LeanRidgelet.affineMackeyLiftFun f x =
    (LeanRidgelet.affineMackeyLiftPhase x)⁻¹ * f ↑x

theorem LeanRidgelet.affineMackeyLiftFun_apply.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {xi : E}
  (f :
    LeanRidgelet.AffineTopologicalMackeyQuotient
        xi →
      ℂ)
  (x :
    AffineEquiv.TopologicalSemidirectProduct
      E) :
  LeanRidgelet.affineMackeyLiftFun f x =
    (LeanRidgelet.affineMackeyLiftPhase
          x)⁻¹ *
      f ↑x

theoremdefined in LeanRidgelet/HA/AffineMackeyLift.lean

complete

theorem LeanRidgelet.norm_affineMackeyLiftFun.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  {xi : E} (hxi : xi ≠ 0)
  (f : LeanRidgelet.AffineTopologicalMackeyQuotient xi → ℂ)
  (x : AffineEquiv.TopologicalSemidirectProduct E) :
  ‖LeanRidgelet.affineMackeyLiftFun f x‖ = ‖f ↑x‖

theorem LeanRidgelet.norm_affineMackeyLiftFun.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {xi : E}
  (hxi : xi ≠ 0)
  (f :
    LeanRidgelet.AffineTopologicalMackeyQuotient
        xi →
      ℂ)
  (x :
    AffineEquiv.TopologicalSemidirectProduct
      E) :
  ‖LeanRidgelet.affineMackeyLiftFun f x‖ =
    ‖f ↑x‖

The lift has the same absolute value as the function it lifts, because the gauge is
unimodular.

theoremdefined in LeanRidgelet/HA/AffineMackeyLift.lean

complete

theorem LeanRidgelet.affineMackeyLiftFun_inv_mul.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  {xi : E} (hxi : xi ≠ 0)
  (f : LeanRidgelet.AffineTopologicalMackeyQuotient xi → ℂ)
  (g x : AffineEquiv.TopologicalSemidirectProduct E) :
  LeanRidgelet.affineMackeyLiftFun f (g⁻¹ * x) =
    (LeanRidgelet.affineMackeyLiftPhase x)⁻¹ *
      (LeanRidgelet.affineTopologicalMackeySectionPhase g ↑x *
        f (g⁻¹ • ↑x))

theorem LeanRidgelet.affineMackeyLiftFun_inv_mul.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {xi : E}
  (hxi : xi ≠ 0)
  (f :
    LeanRidgelet.AffineTopologicalMackeyQuotient
        xi →
      ℂ)
  (g x :
    AffineEquiv.TopologicalSemidirectProduct
      E) :
  LeanRidgelet.affineMackeyLiftFun f
      (g⁻¹ * x) =
    (LeanRidgelet.affineMackeyLiftPhase
          x)⁻¹ *
      (LeanRidgelet.affineTopologicalMackeySectionPhase
          g ↑x *
        f (g⁻¹ • ↑x))

The translation identity that turns smoothing into a convolution: the lift at `g⁻¹ x` is the
induced-action integrand at the coset of `x`, up to the gauge at `x`.

theoremdefined in LeanRidgelet/HA/AffineMackeyLift.lean

complete

theorem LeanRidgelet.measurable_affineMackeyLiftFun.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  {xi : E} (hxi : xi ≠ 0)
  {f : LeanRidgelet.AffineTopologicalMackeyQuotient xi → ℂ}
  (hf : Measurable f) : Measurable (LeanRidgelet.affineMackeyLiftFun f)

theorem LeanRidgelet.measurable_affineMackeyLiftFun.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {xi : E}
  (hxi : xi ≠ 0)
  {f :
    LeanRidgelet.AffineTopologicalMackeyQuotient
        xi →
      ℂ}
  (hf : Measurable f) :
  Measurable
    (LeanRidgelet.affineMackeyLiftFun f)

The lift of a measurable function is measurable: the gauge is continuous and the quotient map is
continuous.

theoremdefined in LeanRidgelet/HA/AffineMackeyLift.lean

complete

theorem LeanRidgelet.affine_map_quotientMk_inv_haar_restrict_le.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E] [MeasurableSpace E]
  [BorelSpace E] {xi : E} (hxi : xi ≠ 0)
  {S : Set (AffineEquiv.TopologicalSemidirectProduct E)}
  (hS : IsCompact S) :
  ∃ C B,
    C ≠ ⊤ ∧
      (LeanRidgelet.affineTopologicalMackeyQuotientMeasure hxi) B ≠ ⊤ ∧
        MeasureTheory.Measure.map (fun x ↦ ↑x⁻¹)
            (MeasureTheory.Measure.haar.restrict S) ≤
          C •
            (LeanRidgelet.affineTopologicalMackeyQuotientMeasure
                  hxi).restrict
              B

theorem LeanRidgelet.affine_map_quotientMk_inv_haar_restrict_le.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E]
  [MeasurableSpace E] [BorelSpace E]
  {xi : E} (hxi : xi ≠ 0)
  {S :
    Set
      (AffineEquiv.TopologicalSemidirectProduct
        E)}
  (hS : IsCompact S) :
  ∃ C B,
    C ≠ ⊤ ∧
      (LeanRidgelet.affineTopologicalMackeyQuotientMeasure
              hxi)
            B ≠
          ⊤ ∧
        MeasureTheory.Measure.map
            (fun x ↦ ↑x⁻¹)
            (MeasureTheory.Measure.haar.restrict
              S) ≤
          C •
            (LeanRidgelet.affineTopologicalMackeyQuotientMeasure
                  hxi).restrict
              B

The image of a compactly restricted Haar measure of the affine group under the inverse coset
map is dominated by a finite multiple of the homogeneous-quotient measure, restricted to a set of
finite measure.

theoremdefined in LeanRidgelet/HA/AffineMackeyLift.lean

complete

theorem LeanRidgelet.locallyIntegrable_affineMackeyLiftFun_inv_of_bound.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] {xi : E}
  (hxi : xi ≠ 0)
  {f : LeanRidgelet.AffineTopologicalMackeyQuotient xi → ℂ}
  (hfmeas : Measurable f)
  (hf :
    MeasureTheory.MemLp f 2
      (LeanRidgelet.affineTopologicalMackeyQuotientMeasure hxi))
  (hbound :
    ∀ (S : Set (AffineEquiv.TopologicalSemidirectProduct E)),
      IsCompact S →
        ∃ C B,
          C ≠ ⊤ ∧
            (LeanRidgelet.affineTopologicalMackeyQuotientMeasure hxi)
                  B ≠
                ⊤ ∧
              MeasureTheory.Measure.map (fun x ↦ ↑x⁻¹)
                  (MeasureTheory.Measure.haar.restrict S) ≤
                C •
                  (LeanRidgelet.affineTopologicalMackeyQuotientMeasure
                        hxi).restrict
                    B) :
  MeasureTheory.LocallyIntegrable
    (fun y ↦ LeanRidgelet.affineMackeyLiftFun f y⁻¹)
    MeasureTheory.Measure.haar

theorem LeanRidgelet.locallyIntegrable_affineMackeyLiftFun_inv_of_bound.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]
  [MeasurableSpace E] [BorelSpace E]
  {xi : E} (hxi : xi ≠ 0)
  {f :
    LeanRidgelet.AffineTopologicalMackeyQuotient
        xi →
      ℂ}
  (hfmeas : Measurable f)
  (hf :
    MeasureTheory.MemLp f 2
      (LeanRidgelet.affineTopologicalMackeyQuotientMeasure
        hxi))
  (hbound :
    ∀
      (S :
        Set
          (AffineEquiv.TopologicalSemidirectProduct
            E)),
      IsCompact S →
        ∃ C B,
          C ≠ ⊤ ∧
            (LeanRidgelet.affineTopologicalMackeyQuotientMeasure
                    hxi)
                  B ≠
                ⊤ ∧
              MeasureTheory.Measure.map
                  (fun x ↦ ↑x⁻¹)
                  (MeasureTheory.Measure.haar.restrict
                    S) ≤
                C •
                  (LeanRidgelet.affineTopologicalMackeyQuotientMeasure
                        hxi).restrict
                    B) :
  MeasureTheory.LocallyIntegrable
    (fun y ↦
      LeanRidgelet.affineMackeyLiftFun f
        y⁻¹)
    MeasureTheory.Measure.haar

The lift of an `L²` class on the affine homogeneous quotient, precomposed with inversion, is
locally integrable on the affine group, provided the image of every compactly restricted Haar
measure under `x ↦ x⁻¹ H` is dominated by a finite multiple of the homogeneous-space measure
restricted to a set of finite measure.

Local compactness reduces the claim to integrability on a compact set. The lift is measurable and
has the same absolute value as the class it lifts, so the lower integral of its norm transports to
the quotient along `x ↦ x⁻¹ H`; the domination hypothesis then bounds it by a finite multiple of
the lower integral of `‖f‖` over a set of finite measure, which is finite because an `L²` function
on a finite measure is integrable.

theoremdefined in LeanRidgelet/HA/AffineMackeyLift.lean

complete

theorem LeanRidgelet.locallyIntegrable_affineMackeyLiftFun_inv.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E] [MeasurableSpace E]
  [BorelSpace E] {xi : E} (hxi : xi ≠ 0)
  {f : LeanRidgelet.AffineTopologicalMackeyQuotient xi → ℂ}
  (hfmeas : Measurable f)
  (hf :
    MeasureTheory.MemLp f 2
      (LeanRidgelet.affineTopologicalMackeyQuotientMeasure hxi)) :
  MeasureTheory.LocallyIntegrable
    (fun y ↦ LeanRidgelet.affineMackeyLiftFun f y⁻¹)
    MeasureTheory.Measure.haar

theorem LeanRidgelet.locallyIntegrable_affineMackeyLiftFun_inv.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E]
  [MeasurableSpace E] [BorelSpace E]
  {xi : E} (hxi : xi ≠ 0)
  {f :
    LeanRidgelet.AffineTopologicalMackeyQuotient
        xi →
      ℂ}
  (hfmeas : Measurable f)
  (hf :
    MeasureTheory.MemLp f 2
      (LeanRidgelet.affineTopologicalMackeyQuotientMeasure
        hxi)) :
  MeasureTheory.LocallyIntegrable
    (fun y ↦
      LeanRidgelet.affineMackeyLiftFun f
        y⁻¹)
    MeasureTheory.Measure.haar

Local integrability of the lift composed with inversion, which is the hypothesis of the
compact-kernel group-convolution continuity theorem.  It combines the quotient-form Haar
pushforward bound with the Cauchy--Schwarz estimate on a finite-measure set.

defdefined in LeanRidgelet/HA/AffineMackeySmoothing.lean

complete

def LeanRidgelet.affineMackeySmoothingKernel.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  (ψ :
    CompactlySupportedContinuousMap
      (AffineEquiv.TopologicalSemidirectProduct E) ℂ) :
  CompactlySupportedContinuousMap
    (AffineEquiv.TopologicalSemidirectProduct E) ℂ

def LeanRidgelet.affineMackeySmoothingKernel.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  (ψ :
    CompactlySupportedContinuousMap
      (AffineEquiv.TopologicalSemidirectProduct
        E)
      ℂ) :
  CompactlySupportedContinuousMap
    (AffineEquiv.TopologicalSemidirectProduct
      E)
    ℂ

Implementation after :=

:= ψ g * ((affineTopologicalJacobian g).sqrt : ℂ)
  continuous_toFun := by
    refine ψ.continuous.mul ?_
    exact Complex.continuous_ofReal.comp
      (NNReal.continuous_coe.comp
        (NNReal.continuous_sqrt.comp continuous_affineTopologicalJacobian))
  hasCompactSupport' := ψ.hasCompactSupport.mul_right

The kernel of the smoothing integral, with the Radon--Nikodym square root absorbed.

theoremdefined in LeanRidgelet/HA/AffineMackeySmoothing.lean

complete

theorem LeanRidgelet.affineMackeySmoothingKernel_apply.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  (ψ :
    CompactlySupportedContinuousMap
      (AffineEquiv.TopologicalSemidirectProduct E) ℂ)
  (g : AffineEquiv.TopologicalSemidirectProduct E) :
  (LeanRidgelet.affineMackeySmoothingKernel ψ) g =
    ψ g * ↑↑(NNReal.sqrt (LeanRidgelet.affineTopologicalJacobian g))

theorem LeanRidgelet.affineMackeySmoothingKernel_apply.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  (ψ :
    CompactlySupportedContinuousMap
      (AffineEquiv.TopologicalSemidirectProduct
        E)
      ℂ)
  (g :
    AffineEquiv.TopologicalSemidirectProduct
      E) :
  (LeanRidgelet.affineMackeySmoothingKernel
        ψ)
      g =
    ψ g *
      ↑↑(NNReal.sqrt
            (LeanRidgelet.affineTopologicalJacobian
              g))

defdefined in LeanRidgelet/HA/AffineMackeySmoothing.lean

complete

def LeanRidgelet.affineMackeySmoothingIntegrand.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  {xi : E}
  (ψ :
    CompactlySupportedContinuousMap
      (AffineEquiv.TopologicalSemidirectProduct E) ℂ)
  (f : LeanRidgelet.AffineTopologicalMackeyQuotient xi → ℂ)
  (g : AffineEquiv.TopologicalSemidirectProduct E)
  (q : LeanRidgelet.AffineTopologicalMackeyQuotient xi) : ℂ

def LeanRidgelet.affineMackeySmoothingIntegrand.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {xi : E}
  (ψ :
    CompactlySupportedContinuousMap
      (AffineEquiv.TopologicalSemidirectProduct
        E)
      ℂ)
  (f :
    LeanRidgelet.AffineTopologicalMackeyQuotient
        xi →
      ℂ)
  (g :
    AffineEquiv.TopologicalSemidirectProduct
      E)
  (q :
    LeanRidgelet.AffineTopologicalMackeyQuotient
      xi) :
  ℂ

Implementation after :=

:=
  ψ g * (affineTopologicalMackeySectionPhase g q *
    (((affineTopologicalJacobian g).sqrt : ℂ) * f (g⁻¹ • q)))

The integrand of the smoothing integral, as a function of the group element and of the point of
the homogeneous quotient.  Each slice in the group variable is the explicit almost-everywhere
representative of the induced action, scaled by the smoothing kernel.

theoremdefined in LeanRidgelet/HA/AffineMackeySmoothing.lean

complete

theorem LeanRidgelet.measurable_uncurry_affineMackeySmoothingIntegrand.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {xi : E} (hxi : xi ≠ 0)
  (ψ :
    CompactlySupportedContinuousMap
      (AffineEquiv.TopologicalSemidirectProduct E) ℂ)
  {f : LeanRidgelet.AffineTopologicalMackeyQuotient xi → ℂ}
  (hfmeas : Measurable f) :
  Measurable
    (Function.uncurry (LeanRidgelet.affineMackeySmoothingIntegrand ψ f))

theorem LeanRidgelet.measurable_uncurry_affineMackeySmoothingIntegrand.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {xi : E}
  (hxi : xi ≠ 0)
  (ψ :
    CompactlySupportedContinuousMap
      (AffineEquiv.TopologicalSemidirectProduct
        E)
      ℂ)
  {f :
    LeanRidgelet.AffineTopologicalMackeyQuotient
        xi →
      ℂ}
  (hfmeas : Measurable f) :
  Measurable
    (Function.uncurry
      (LeanRidgelet.affineMackeySmoothingIntegrand
        ψ f))

The smoothing integrand is jointly measurable: every factor but `f` is jointly continuous, and
`f` is composed with the continuous group action on the homogeneous quotient.

theoremdefined in LeanRidgelet/HA/AffineMackeySmoothing.lean

complete

theorem LeanRidgelet.affineMackeySmoothingIntegrand_ae_eq.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  [Nontrivial E] [MeasurableSpace E] [BorelSpace E] {xi : E}
  (hxi : xi ≠ 0)
  (ψ :
    CompactlySupportedContinuousMap
      (AffineEquiv.TopologicalSemidirectProduct E) ℂ)
  {f : LeanRidgelet.AffineTopologicalMackeyQuotient xi → ℂ}
  {F :
    ↥(MeasureTheory.Lp ℂ 2
        (LeanRidgelet.affineTopologicalMackeyQuotientMeasure hxi))}
  (hfF :
    f =ᵐ[LeanRidgelet.affineTopologicalMackeyQuotientMeasure hxi] ↑↑F)
  (g : AffineEquiv.TopologicalSemidirectProduct E) :
  LeanRidgelet.affineMackeySmoothingIntegrand ψ f
      g =ᵐ[LeanRidgelet.affineTopologicalMackeyQuotientMeasure hxi]
    fun q ↦
    ψ g •
      ↑↑(↑((LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation
                    hxi)
                  g)
              F)
        q

theorem LeanRidgelet.affineMackeySmoothingIntegrand_ae_eq.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E]
  [MeasurableSpace E] [BorelSpace E]
  {xi : E} (hxi : xi ≠ 0)
  (ψ :
    CompactlySupportedContinuousMap
      (AffineEquiv.TopologicalSemidirectProduct
        E)
      ℂ)
  {f :
    LeanRidgelet.AffineTopologicalMackeyQuotient
        xi →
      ℂ}
  {F :
    ↥(MeasureTheory.Lp ℂ 2
        (LeanRidgelet.affineTopologicalMackeyQuotientMeasure
          hxi))}
  (hfF :
    f =ᵐ[LeanRidgelet.affineTopologicalMackeyQuotientMeasure
        hxi]
      ↑↑F)
  (g :
    AffineEquiv.TopologicalSemidirectProduct
      E) :
  LeanRidgelet.affineMackeySmoothingIntegrand
      ψ f
      g =ᵐ[LeanRidgelet.affineTopologicalMackeyQuotientMeasure
      hxi]
    fun q ↦
    ψ g •
      ↑↑(↑((LeanRidgelet.affineTopologicalMackeySectionInducedLpUnitaryRepresentation
                    hxi)
                  g)
              F)
        q

Each slice of the smoothing integrand is almost everywhere the value of the kernel times the
normalized-section induced action applied to the `L²` class that `f` represents.

theoremdefined in LeanRidgelet/HA/AffineMackeySmoothing.lean

complete

theorem LeanRidgelet.integrable_uncurry_affineMackeySmoothingIntegrand.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E] [MeasurableSpace E]
  [BorelSpace E] {xi : E} (hxi : xi ≠ 0)
  (ψ :
    CompactlySupportedContinuousMap
      (AffineEquiv.TopologicalSemidirectProduct E) ℂ)
  {f : LeanRidgelet.AffineTopologicalMackeyQuotient xi → ℂ}
  (hfmeas : Measurable f)
  {F :
    ↥(MeasureTheory.Lp ℂ 2
        (LeanRidgelet.affineTopologicalMackeyQuotientMeasure hxi))}
  (hfF :
    f =ᵐ[LeanRidgelet.affineTopologicalMackeyQuotientMeasure hxi] ↑↑F)
  {s : Set (LeanRidgelet.AffineTopologicalMackeyQuotient xi)}
  (hs : MeasurableSet s)
  (hfin :
    (LeanRidgelet.affineTopologicalMackeyQuotientMeasure hxi) s ≠ ⊤) :
  MeasureTheory.Integrable
    (Function.uncurry (LeanRidgelet.affineMackeySmoothingIntegrand ψ f))
    (MeasureTheory.Measure.haar.prod
      ((LeanRidgelet.affineTopologicalMackeyQuotientMeasure
            hxi).restrict
        s))

theorem LeanRidgelet.integrable_uncurry_affineMackeySmoothingIntegrand.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [Nontrivial E]
  [MeasurableSpace E] [BorelSpace E]
  {xi : E} (hxi : xi ≠ 0)
  (ψ :
    CompactlySupportedContinuousMap
      (AffineEquiv.TopologicalSemidirectProduct
        E)
      ℂ)
  {f :
    LeanRidgelet.AffineTopologicalMackeyQuotient
        xi →
      ℂ}
  (hfmeas : Measurable f)
  {F :
    ↥(MeasureTheory.Lp ℂ 2
        (LeanRidgelet.affineTopologicalMackeyQuotientMeasure
          hxi))}
  (hfF :
    f =ᵐ[LeanRidgelet.affineTopologicalMackeyQuotientMeasure
        hxi]
      ↑↑F)
  {s :
    Set
      (LeanRidgelet.AffineTopologicalMackeyQuotient
        xi)}
  (hs : MeasurableSet s)
  (hfin :
    (LeanRidgelet.affineTopologicalMackeyQuotientMeasure
          hxi)
        s ≠
      ⊤) :
  MeasureTheory.Integrable
    (Function.uncurry
      (LeanRidgelet.affineMackeySmoothingIntegrand
        ψ f))
    (MeasureTheory.Measure.haar.prod
      ((LeanRidgelet.affineTopologicalMackeyQuotientMeasure
            hxi).restrict
        s))

The smoothing integrand is integrable on the product of the affine group with a finite-measure
part of the homogeneous quotient.  Its slices are `L²` classes on the quotient, hence integrable on
a set of finite measure, and Hölder's inequality bounds their `L¹` norms uniformly in the group
variable because the induced action is unitary; the smoothing kernel then supplies a compactly
supported dominating function on the group.

defdefined in LeanRidgelet/HA/AffineMackeySmoothing.lean

complete

def LeanRidgelet.affineMackeySmoothingIntegral.{u_1} {E : Type u_1}
  [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
  {xi : E}
  (ψ :
    CompactlySupportedContinuousMap
      (AffineEquiv.TopologicalSemidirectProduct E) ℂ)
  (f : LeanRidgelet.AffineTopologicalMackeyQuotient xi → ℂ)
  (q : LeanRidgelet.AffineTopologicalMackeyQuotient xi) : ℂ

def LeanRidgelet.affineMackeySmoothingIntegral.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {xi : E}
  (ψ :
    CompactlySupportedContinuousMap
      (AffineEquiv.TopologicalSemidirectProduct
        E)
      ℂ)
  (f :
    LeanRidgelet.AffineTopologicalMackeyQuotient
        xi →
      ℂ)
  (q :
    LeanRidgelet.AffineTopologicalMackeyQuotient
      xi) :
  ℂ

Implementation after :=

:=
  ∫ g, affineMackeySmoothingIntegrand ψ f g q ∂Measure.haar

The pointwise smoothing integral on the homogeneous quotient.

theoremdefined in LeanRidgelet/HA/AffineMackeySmoothing.lean

complete

theorem LeanRidgelet.affineMackeySmoothingIntegral_quotientMk.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {xi : E} (hxi : xi ≠ 0)
  (ψ :
    CompactlySupportedContinuousMap
      (AffineEquiv.TopologicalSemidirectProduct E) ℂ)
  (f : LeanRidgelet.AffineTopologicalMackeyQuotient xi → ℂ)
  (x : AffineEquiv.TopologicalSemidirectProduct E) :
  LeanRidgelet.affineMackeySmoothingIntegral ψ f ↑x =
    LeanRidgelet.affineMackeyLiftPhase x *
      ∫ (y : AffineEquiv.TopologicalSemidirectProduct E),
        LeanRidgelet.affineMackeyLiftFun f y⁻¹ *
          (LeanRidgelet.affineMackeySmoothingKernel ψ)
            (x * y) ∂MeasureTheory.Measure.haar

theorem LeanRidgelet.affineMackeySmoothingIntegral_quotientMk.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {xi : E}
  (hxi : xi ≠ 0)
  (ψ :
    CompactlySupportedContinuousMap
      (AffineEquiv.TopologicalSemidirectProduct
        E)
      ℂ)
  (f :
    LeanRidgelet.AffineTopologicalMackeyQuotient
        xi →
      ℂ)
  (x :
    AffineEquiv.TopologicalSemidirectProduct
      E) :
  LeanRidgelet.affineMackeySmoothingIntegral
      ψ f ↑x =
    LeanRidgelet.affineMackeyLiftPhase x *
      ∫ (y :
        AffineEquiv.TopologicalSemidirectProduct
          E),
        LeanRidgelet.affineMackeyLiftFun f
            y⁻¹ *
          (LeanRidgelet.affineMackeySmoothingKernel
              ψ)
            (x *
              y) ∂MeasureTheory.Measure.haar

The smoothing integral along the quotient map is the group convolution of the smoothing kernel
with the lift, corrected by the gauge.  This is the identity that makes the convolution continuity
theorem applicable.

theoremdefined in LeanRidgelet/HA/AffineMackeySmoothing.lean

complete

theorem LeanRidgelet.continuous_affineMackeySmoothingIntegral.{u_1}
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {xi : E} (hxi : xi ≠ 0)
  (ψ :
    CompactlySupportedContinuousMap
      (AffineEquiv.TopologicalSemidirectProduct E) ℂ)
  {f : LeanRidgelet.AffineTopologicalMackeyQuotient xi → ℂ}
  (hloc :
    MeasureTheory.LocallyIntegrable
      (fun y ↦ LeanRidgelet.affineMackeyLiftFun f y⁻¹)
      MeasureTheory.Measure.haar) :
  Continuous (LeanRidgelet.affineMackeySmoothingIntegral ψ f)

theorem LeanRidgelet.continuous_affineMackeySmoothingIntegral.{u_1}
  {E : Type u_1} [NormedAddCommGroup E]
  [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] {xi : E}
  (hxi : xi ≠ 0)
  (ψ :
    CompactlySupportedContinuousMap
      (AffineEquiv.TopologicalSemidirectProduct
        E)
      ℂ)
  {f :
    LeanRidgelet.AffineTopologicalMackeyQuotient
        xi →
      ℂ}
  (hloc :
    MeasureTheory.LocallyIntegrable
      (fun y ↦
        LeanRidgelet.affineMackeyLiftFun f
          y⁻¹)
      MeasureTheory.Measure.haar) :
  Continuous
    (LeanRidgelet.affineMackeySmoothingIntegral
      ψ f)

Continuity of the smoothing integral on the homogeneous quotient, given local integrability of
the lift.  The convolution is continuous by the compact-kernel group-convolution theorem, the gauge
is continuous, and the quotient map is a topological quotient map.

The gauge P is unimodular, nowhere zero, and continuous — although the normalized section is only set-theoretic, its character equals the explicit Fourier phase, and that phase is continuous in both arguments. The cocycle law of the section phase gives P(x)=\mathrm{phase}(g,xH)\,P(g^{-1}x), hence the translation identity F(g^{-1}x)=P(x)^{-1}\bigl(\mathrm{phase}(g,xH)\,f(g^{-1}\cdot xH)\bigr), whose right-hand side is exactly the integrand of the induced action at the coset of x, up to the factor P(x)^{-1} that does not depend on the integration variable. Convolving with a compactly supported kernel therefore reproduces the smoothed vector along the quotient map, and continuity of the convolution transfers to the quotient because the quotient map is a topological quotient map. Absolute values of the lift are those of the original function, which is what the local-integrability estimate consumes.

The estimate and the lift combine into continuity of the smoothing integral. Transporting the frequency-space Haar bound along the quotient-orbit homeomorphism gives the same bound on the homogeneous quotient, and Cauchy--Schwarz on a finite-measure set turns it into local integrability of the lift composed with inversion — exactly the hypothesis of the compact-kernel group-convolution theorem. Along the quotient map the smoothing integral is that convolution, of the compactly supported kernel ψ\cdot\sqrt{\det} with the lift, corrected by the continuous unimodular gauge; hence it is continuous on the group, and it descends because the quotient map of a topological group by a subgroup is a quotient map.

What is left for Lemma 6.29 is to know that this pointwise integral is the representative of the Bochner-integrated vector, and that is a slice-integrability question. Each slice of the smoothing integrand in the group variable is, almost everywhere, the value of the kernel times the induced action applied to the class of the data, so all slices have the same L^2 norm; on a set of finite measure Hölder's inequality turns that into a uniform bound on their L^1 norms, and the compactly supported kernel dominates the group variable. The integrand is jointly measurable because the section phase is jointly continuous in the group element and the coset. The family is therefore integrable on the product of the group with every finite-measure part of the quotient, which is exactly the hypothesis of the slice form of the pointwise-representative theorem: a measurable representative of the data yields a continuous representative of the smoothed vector, and the Lemma 6.29 root is closed.