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Multiplicity One Theorems
D. Gourevitch
http://www.wisdom.weizmann.ac.il/ ∼ dimagur/
D. Gourevitch Multiplicity One Theorems
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Formulation
Let F be a local field of characteristic zero.
Theorem (Aizenbud-Gourevitch-Rallis-Schiffmann-Sun-Zhu)
Every GLn(F )-invariant distribution on GLn+1(F ) istransposition invariant.
It has the following corollary in representation theory.
TheoremLet π be an irreducible admissible representation of GLn+1(F )and τ be an irreducible admissible representation of GLn(F ).Then
dim HomGLn(F )(π, τ) ≤ 1.
Similar theorems hold for orthogonal and unitary groups.
D. Gourevitch Multiplicity One Theorems
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Formulation
Let F be a local field of characteristic zero.
Theorem (Aizenbud-Gourevitch-Rallis-Schiffmann-Sun-Zhu)
Every GLn(F )-invariant distribution on GLn+1(F ) istransposition invariant.
It has the following corollary in representation theory.
TheoremLet π be an irreducible admissible representation of GLn+1(F )and τ be an irreducible admissible representation of GLn(F ).Then
dim HomGLn(F )(π, τ) ≤ 1.
Similar theorems hold for orthogonal and unitary groups.
D. Gourevitch Multiplicity One Theorems
![Page 4: Multiplicity One Theorems](https://reader030.vdocuments.site/reader030/viewer/2022012616/619e7283b52c3b6ac7137851/html5/thumbnails/4.jpg)
Formulation
Let F be a local field of characteristic zero.
Theorem (Aizenbud-Gourevitch-Rallis-Schiffmann-Sun-Zhu)
Every GLn(F )-invariant distribution on GLn+1(F ) istransposition invariant.
It has the following corollary in representation theory.
TheoremLet π be an irreducible admissible representation of GLn+1(F )and τ be an irreducible admissible representation of GLn(F ).Then
dim HomGLn(F )(π, τ) ≤ 1.
Similar theorems hold for orthogonal and unitary groups.
D. Gourevitch Multiplicity One Theorems
![Page 5: Multiplicity One Theorems](https://reader030.vdocuments.site/reader030/viewer/2022012616/619e7283b52c3b6ac7137851/html5/thumbnails/5.jpg)
Distributions
NotationLet M be a smooth manifold. We denote by C∞c (M) the spaceof smooth compactly supported functions on M. We willconsider the space (C∞c (M))∗ of distributions on M. Sometimeswe will also consider the space S∗(M) of Schwartz distributionson M.
DefinitionAn `-space is a Hausdorff locally compact totally disconnectedtopological space. For an `-space X we denote by S(X ) thespace of compactly supported locally constant functions on X .We let S∗(X ) := S(X )∗ be the space of distributions on X .
D. Gourevitch Multiplicity One Theorems
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Distributions
NotationLet M be a smooth manifold. We denote by C∞c (M) the spaceof smooth compactly supported functions on M. We willconsider the space (C∞c (M))∗ of distributions on M. Sometimeswe will also consider the space S∗(M) of Schwartz distributionson M.
DefinitionAn `-space is a Hausdorff locally compact totally disconnectedtopological space. For an `-space X we denote by S(X ) thespace of compactly supported locally constant functions on X .We let S∗(X ) := S(X )∗ be the space of distributions on X .
D. Gourevitch Multiplicity One Theorems
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G̃ := GLn(F ) o {1, σ}Define a character χ of G̃ by χ(GLn(F )) = {1},χ(G̃ −GLn(F )) = {−1}.
Equivalent formulation:
Theorem
S∗(GLn+1(F ))G̃,χ = 0.
D. Gourevitch Multiplicity One Theorems
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G̃ := GLn(F ) o {1, σ}Define a character χ of G̃ by χ(GLn(F )) = {1},χ(G̃ −GLn(F )) = {−1}.
Equivalent formulation:
Theorem
S∗(GLn+1(F ))G̃,χ = 0.
D. Gourevitch Multiplicity One Theorems
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Equivalent formulation:
Theorem
S∗(gln+1(F ))G̃,χ = 0.
g(
An×n vn×1φ1×n λ
)g−1 =
(gAg−1 gv(g∗)−1φ λ
)and
(A vφ λ
)t
=
(At φt
v t λ
)V := F n
X := sl(V )× V × V ∗
G̃ acts on X byg(A, v , φ) = (gAg−1,gv , (g∗)−1φ)σ(A, v , φ) = (At , φt , v t).
Equivalent formulation:
Theorem
S∗(X )G̃,χ = 0.
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Equivalent formulation:
Theorem
S∗(gln+1(F ))G̃,χ = 0.
g(
An×n vn×1φ1×n λ
)g−1 =
(gAg−1 gv(g∗)−1φ λ
)and
(A vφ λ
)t
=
(At φt
v t λ
)
V := F n
X := sl(V )× V × V ∗
G̃ acts on X byg(A, v , φ) = (gAg−1,gv , (g∗)−1φ)σ(A, v , φ) = (At , φt , v t).
Equivalent formulation:
Theorem
S∗(X )G̃,χ = 0.
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Equivalent formulation:
Theorem
S∗(gln+1(F ))G̃,χ = 0.
g(
An×n vn×1φ1×n λ
)g−1 =
(gAg−1 gv(g∗)−1φ λ
)and
(A vφ λ
)t
=
(At φt
v t λ
)V := F n
X := sl(V )× V × V ∗
G̃ acts on X byg(A, v , φ) = (gAg−1,gv , (g∗)−1φ)σ(A, v , φ) = (At , φt , v t).
Equivalent formulation:
Theorem
S∗(X )G̃,χ = 0.
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Equivalent formulation:
Theorem
S∗(gln+1(F ))G̃,χ = 0.
g(
An×n vn×1φ1×n λ
)g−1 =
(gAg−1 gv(g∗)−1φ λ
)and
(A vφ λ
)t
=
(At φt
v t λ
)V := F n
X := sl(V )× V × V ∗
G̃ acts on X byg(A, v , φ) = (gAg−1,gv , (g∗)−1φ)σ(A, v , φ) = (At , φt , v t).
Equivalent formulation:
Theorem
S∗(X )G̃,χ = 0.
![Page 13: Multiplicity One Theorems](https://reader030.vdocuments.site/reader030/viewer/2022012616/619e7283b52c3b6ac7137851/html5/thumbnails/13.jpg)
Equivalent formulation:
Theorem
S∗(gln+1(F ))G̃,χ = 0.
g(
An×n vn×1φ1×n λ
)g−1 =
(gAg−1 gv(g∗)−1φ λ
)and
(A vφ λ
)t
=
(At φt
v t λ
)V := F n
X := sl(V )× V × V ∗
G̃ acts on X byg(A, v , φ) = (gAg−1,gv , (g∗)−1φ)σ(A, v , φ) = (At , φt , v t).
Equivalent formulation:
Theorem
S∗(X )G̃,χ = 0.
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First tool: Stratification
Setting
A group G acts on a space X, and χ is a character of G. Wewant to show S∗(X )G,χ = 0.
PropositionLet U ⊂ X be an open G-invariant subset and Z := X − U.Suppose that S∗(U)G,χ = 0 and S∗X (Z )G,χ = 0. ThenS∗(X )G,χ = 0.
Proof.
0→ S∗X (Z )G,χ → S∗(X )G,χ → S∗(U)G,χ.
For `-spaces, S∗X (Z )G,χ ∼= S∗(Z )G,χ.For smooth manifolds, there is a slightly more complicatedstatement which takes into account transversal derivatives.
D. Gourevitch Multiplicity One Theorems
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First tool: Stratification
Setting
A group G acts on a space X, and χ is a character of G. Wewant to show S∗(X )G,χ = 0.
PropositionLet U ⊂ X be an open G-invariant subset and Z := X − U.Suppose that S∗(U)G,χ = 0 and S∗X (Z )G,χ = 0. ThenS∗(X )G,χ = 0.
Proof.
0→ S∗X (Z )G,χ → S∗(X )G,χ → S∗(U)G,χ.
For `-spaces, S∗X (Z )G,χ ∼= S∗(Z )G,χ.For smooth manifolds, there is a slightly more complicatedstatement which takes into account transversal derivatives.
D. Gourevitch Multiplicity One Theorems
![Page 16: Multiplicity One Theorems](https://reader030.vdocuments.site/reader030/viewer/2022012616/619e7283b52c3b6ac7137851/html5/thumbnails/16.jpg)
First tool: Stratification
Setting
A group G acts on a space X, and χ is a character of G. Wewant to show S∗(X )G,χ = 0.
PropositionLet U ⊂ X be an open G-invariant subset and Z := X − U.Suppose that S∗(U)G,χ = 0 and S∗X (Z )G,χ = 0. ThenS∗(X )G,χ = 0.
Proof.
0→ S∗X (Z )G,χ → S∗(X )G,χ → S∗(U)G,χ.
For `-spaces, S∗X (Z )G,χ ∼= S∗(Z )G,χ.For smooth manifolds, there is a slightly more complicatedstatement which takes into account transversal derivatives.
D. Gourevitch Multiplicity One Theorems
![Page 17: Multiplicity One Theorems](https://reader030.vdocuments.site/reader030/viewer/2022012616/619e7283b52c3b6ac7137851/html5/thumbnails/17.jpg)
First tool: Stratification
Setting
A group G acts on a space X, and χ is a character of G. Wewant to show S∗(X )G,χ = 0.
PropositionLet U ⊂ X be an open G-invariant subset and Z := X − U.Suppose that S∗(U)G,χ = 0 and S∗X (Z )G,χ = 0. ThenS∗(X )G,χ = 0.
Proof.
0→ S∗X (Z )G,χ → S∗(X )G,χ → S∗(U)G,χ.
For `-spaces, S∗X (Z )G,χ ∼= S∗(Z )G,χ.
For smooth manifolds, there is a slightly more complicatedstatement which takes into account transversal derivatives.
D. Gourevitch Multiplicity One Theorems
![Page 18: Multiplicity One Theorems](https://reader030.vdocuments.site/reader030/viewer/2022012616/619e7283b52c3b6ac7137851/html5/thumbnails/18.jpg)
First tool: Stratification
Setting
A group G acts on a space X, and χ is a character of G. Wewant to show S∗(X )G,χ = 0.
PropositionLet U ⊂ X be an open G-invariant subset and Z := X − U.Suppose that S∗(U)G,χ = 0 and S∗X (Z )G,χ = 0. ThenS∗(X )G,χ = 0.
Proof.
0→ S∗X (Z )G,χ → S∗(X )G,χ → S∗(U)G,χ.
For `-spaces, S∗X (Z )G,χ ∼= S∗(Z )G,χ.For smooth manifolds, there is a slightly more complicatedstatement which takes into account transversal derivatives.
D. Gourevitch Multiplicity One Theorems
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Frobenius descent
Xz
��
// X
��z // Z
Theorem (Bernstein, Baruch, ...)
Let ψ : X → Z be a map.Let G act on X and Z such that ψ(gx) = gψ(x).Suppose that the action of G on Z is transitive.Suppose that both G and StabG(z) are unimodular. Then
S∗(X )G,χ ∼= S∗(Xz)StabG(z),χ.
D. Gourevitch Multiplicity One Theorems
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Generalized Harish-Chandra descent
TheoremLet a reductive group G act on a smooth affine algebraic varietyX. Let χ be a character of G. Suppose that for any a ∈ X s.t.the orbit Ga is closed we have
S∗(NXGa,a)
Ga,χ = 0.
Then S∗(X )G,χ = 0.
D. Gourevitch Multiplicity One Theorems
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Fourier transform
Let V be a finite dimensional vector space over F and Q be anon-degenerate quadratic form on V . Let ξ̂ denote the Fouriertransform of ξ defined using Q.
Proposition
Let G act on V linearly and preserving Q. Let ξ ∈ S∗(V )G,χ.Then ξ̂ ∈ S∗(V )G,χ.
D. Gourevitch Multiplicity One Theorems
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Fourier transform and homogeneity
We call a distribution ξ ∈ S∗(V ) abs-homogeneous ofdegree d if for any t ∈ F×,
ht(ξ) = u(t)|t |dξ,where ht denotes the homothety action on distributionsand u is some unitary character of F×.
Theorem (Jacquet, Rallis, Schiffmann,...)
Assume F is non-archimedean. Let ξ ∈ S∗V (Z (Q)) be s.t.ξ̂ ∈ S∗V (Z (Q)). Then ξ is abs-homogeneous of degree 1
2dimV.
Theorem (archimedean homogeneity)
Let F be any local field. Let L ⊂ S∗V (Z (Q)) be a non-zero linearsubspace s. t. ∀ξ ∈ L we have ξ̂ ∈ L and Qξ ∈ L.Then there exists a non-zero distribution ξ ∈ L which isabs-homogeneous of degree 1
2dimV or of degree 12dimV + 1.
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Fourier transform and homogeneity
We call a distribution ξ ∈ S∗(V ) abs-homogeneous ofdegree d if for any t ∈ F×,
ht(ξ) = u(t)|t |dξ,where ht denotes the homothety action on distributionsand u is some unitary character of F×.
Theorem (Jacquet, Rallis, Schiffmann,...)
Assume F is non-archimedean. Let ξ ∈ S∗V (Z (Q)) be s.t.ξ̂ ∈ S∗V (Z (Q)). Then ξ is abs-homogeneous of degree 1
2dimV.
Theorem (archimedean homogeneity)
Let F be any local field. Let L ⊂ S∗V (Z (Q)) be a non-zero linearsubspace s. t. ∀ξ ∈ L we have ξ̂ ∈ L and Qξ ∈ L.Then there exists a non-zero distribution ξ ∈ L which isabs-homogeneous of degree 1
2dimV or of degree 12dimV + 1.
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Fourier transform and homogeneity
We call a distribution ξ ∈ S∗(V ) abs-homogeneous ofdegree d if for any t ∈ F×,
ht(ξ) = u(t)|t |dξ,where ht denotes the homothety action on distributionsand u is some unitary character of F×.
Theorem (Jacquet, Rallis, Schiffmann,...)
Assume F is non-archimedean. Let ξ ∈ S∗V (Z (Q)) be s.t.ξ̂ ∈ S∗V (Z (Q)). Then ξ is abs-homogeneous of degree 1
2dimV.
Theorem (archimedean homogeneity)
Let F be any local field. Let L ⊂ S∗V (Z (Q)) be a non-zero linearsubspace s. t. ∀ξ ∈ L we have ξ̂ ∈ L and Qξ ∈ L.Then there exists a non-zero distribution ξ ∈ L which isabs-homogeneous of degree 1
2dimV or of degree 12dimV + 1.
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Singular Support and Wave Front Set
To a distribution ξ on X one assigns two subsets of T ∗X .
Singular Support Wave front set(=Characteristic variety)
Defined using D-modules Defined using Fourier transformAvailable only in the Available in both casesArchimedean case
In the non-Archimedean case we define the singular support tobe the Zariski closure of the wave front set.
D. Gourevitch Multiplicity One Theorems
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Singular Support and Wave Front Set
To a distribution ξ on X one assigns two subsets of T ∗X .Singular Support Wave front set
(=Characteristic variety)Defined using D-modules Defined using Fourier transform
Available only in the Available in both casesArchimedean case
In the non-Archimedean case we define the singular support tobe the Zariski closure of the wave front set.
D. Gourevitch Multiplicity One Theorems
![Page 27: Multiplicity One Theorems](https://reader030.vdocuments.site/reader030/viewer/2022012616/619e7283b52c3b6ac7137851/html5/thumbnails/27.jpg)
Singular Support and Wave Front Set
To a distribution ξ on X one assigns two subsets of T ∗X .Singular Support Wave front set
(=Characteristic variety)Defined using D-modules Defined using Fourier transform
Available only in the Available in both casesArchimedean case
In the non-Archimedean case we define the singular support tobe the Zariski closure of the wave front set.
D. Gourevitch Multiplicity One Theorems
![Page 28: Multiplicity One Theorems](https://reader030.vdocuments.site/reader030/viewer/2022012616/619e7283b52c3b6ac7137851/html5/thumbnails/28.jpg)
Properties and the Integrability Theorem
Let X be a smooth algebraic variety.Let ξ ∈ S∗(X ). Then Supp(ξ)Zar = pX (SS(ξ)), wherepX : T ∗X → X is the projection.
Let an algebraic group G act on X . Let ξ ∈ S∗(X )G,χ. Then
SS(ξ) ⊂ {(x , φ) ∈ T ∗X | ∀α ∈ g φ(α(x)) = 0}.
Let V be a linear space. Let Z ⊂ V ∗ be a closedsubvariety, invariant with respect to homotheties in V . Letξ ∈ S∗(V ). Suppose that Supp(ξ̂) ⊂ Z . ThenSS(ξ) ⊂ V × Z .Integrability theorem:Let ξ ∈ S∗(X ). Then SS(ξ) is (weakly) coisotropic.
D. Gourevitch Multiplicity One Theorems
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Properties and the Integrability Theorem
Let X be a smooth algebraic variety.Let ξ ∈ S∗(X ). Then Supp(ξ)Zar = pX (SS(ξ)), wherepX : T ∗X → X is the projection.Let an algebraic group G act on X . Let ξ ∈ S∗(X )G,χ. Then
SS(ξ) ⊂ {(x , φ) ∈ T ∗X | ∀α ∈ g φ(α(x)) = 0}.
Let V be a linear space. Let Z ⊂ V ∗ be a closedsubvariety, invariant with respect to homotheties in V . Letξ ∈ S∗(V ). Suppose that Supp(ξ̂) ⊂ Z . ThenSS(ξ) ⊂ V × Z .Integrability theorem:Let ξ ∈ S∗(X ). Then SS(ξ) is (weakly) coisotropic.
D. Gourevitch Multiplicity One Theorems
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Properties and the Integrability Theorem
Let X be a smooth algebraic variety.Let ξ ∈ S∗(X ). Then Supp(ξ)Zar = pX (SS(ξ)), wherepX : T ∗X → X is the projection.Let an algebraic group G act on X . Let ξ ∈ S∗(X )G,χ. Then
SS(ξ) ⊂ {(x , φ) ∈ T ∗X | ∀α ∈ g φ(α(x)) = 0}.
Let V be a linear space. Let Z ⊂ V ∗ be a closedsubvariety, invariant with respect to homotheties in V . Letξ ∈ S∗(V ). Suppose that Supp(ξ̂) ⊂ Z . ThenSS(ξ) ⊂ V × Z .
Integrability theorem:Let ξ ∈ S∗(X ). Then SS(ξ) is (weakly) coisotropic.
D. Gourevitch Multiplicity One Theorems
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Properties and the Integrability Theorem
Let X be a smooth algebraic variety.Let ξ ∈ S∗(X ). Then Supp(ξ)Zar = pX (SS(ξ)), wherepX : T ∗X → X is the projection.Let an algebraic group G act on X . Let ξ ∈ S∗(X )G,χ. Then
SS(ξ) ⊂ {(x , φ) ∈ T ∗X | ∀α ∈ g φ(α(x)) = 0}.
Let V be a linear space. Let Z ⊂ V ∗ be a closedsubvariety, invariant with respect to homotheties in V . Letξ ∈ S∗(V ). Suppose that Supp(ξ̂) ⊂ Z . ThenSS(ξ) ⊂ V × Z .Integrability theorem:Let ξ ∈ S∗(X ). Then SS(ξ) is (weakly) coisotropic.
D. Gourevitch Multiplicity One Theorems
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Coisotropic varieties
DefinitionLet M be a smooth algebraic variety and ω be a symplecticform on it. Let Z ⊂ M be an algebraic subvariety. We call itM-coisotropic if the following equivalent conditions holds.
At every smooth point z ∈ Z we have TzZ ⊃ (TzZ )⊥. Here,(TzZ )⊥ denotes the orthogonal complement to TzZ in TzMwith respect to ω.The ideal sheaf of regular functions that vanish on Z isclosed under Poisson bracket.
If there is no ambiguity, we will call Z a coisotropic variety.
Every non-empty coisotropic subvariety of M hasdimension at least dim M
2 .
D. Gourevitch Multiplicity One Theorems
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Coisotropic varieties
DefinitionLet M be a smooth algebraic variety and ω be a symplecticform on it. Let Z ⊂ M be an algebraic subvariety. We call itM-coisotropic if the following equivalent conditions holds.
At every smooth point z ∈ Z we have TzZ ⊃ (TzZ )⊥. Here,(TzZ )⊥ denotes the orthogonal complement to TzZ in TzMwith respect to ω.The ideal sheaf of regular functions that vanish on Z isclosed under Poisson bracket.
If there is no ambiguity, we will call Z a coisotropic variety.
Every non-empty coisotropic subvariety of M hasdimension at least dim M
2 .
D. Gourevitch Multiplicity One Theorems
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Weakly coisotropic varieties
DefinitionLet X be a smooth algebraic variety. Let Z ⊂ T ∗X be analgebraic subvariety. We call it T ∗X -weakly coisotropic if oneof the following equivalent conditions holds.
For a generic smooth point a ∈ pX (Z ) and for a genericsmooth point y ∈ p−1
X (a) ∩ Z we haveCNX
pX (Z ),a ⊂ Ty (p−1X (a) ∩ Z ).
For any smooth point a ∈ pX (Z ) the fiber p−1X (a) ∩ Z is
locally invariant with respect to shifts by CNXpX (Z ),a.
Every non-empty weakly coisotropic subvariety of T ∗X hasdimension at least dim X .
D. Gourevitch Multiplicity One Theorems
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Weakly coisotropic varieties
DefinitionLet X be a smooth algebraic variety. Let Z ⊂ T ∗X be analgebraic subvariety. We call it T ∗X -weakly coisotropic if oneof the following equivalent conditions holds.
For a generic smooth point a ∈ pX (Z ) and for a genericsmooth point y ∈ p−1
X (a) ∩ Z we haveCNX
pX (Z ),a ⊂ Ty (p−1X (a) ∩ Z ).
For any smooth point a ∈ pX (Z ) the fiber p−1X (a) ∩ Z is
locally invariant with respect to shifts by CNXpX (Z ),a.
Every non-empty weakly coisotropic subvariety of T ∗X hasdimension at least dim X .
D. Gourevitch Multiplicity One Theorems
![Page 36: Multiplicity One Theorems](https://reader030.vdocuments.site/reader030/viewer/2022012616/619e7283b52c3b6ac7137851/html5/thumbnails/36.jpg)
DefinitionLet X be a smooth algebraic variety. Let Z ⊂ X be a smoothsubvariety and R ⊂ T ∗X be any subvariety. We define therestriction R|Z ⊂ T ∗Z of R to Z by
R|Z := q(p−1X (Z ) ∩ R),
where q : p−1X (Z )→ T ∗Z is the projection.
T ∗X ⊃ p−1Y (Z ) � T ∗Z
Lemma
Let X be a smooth algebraic variety. Let Z ⊂ X be a smoothsubvariety. Let R ⊂ T ∗X be a (weakly) coisotropic variety.Then, under some transversality assumption, R|Z ⊂ T ∗Z is a(weakly) coisotropic variety.
![Page 37: Multiplicity One Theorems](https://reader030.vdocuments.site/reader030/viewer/2022012616/619e7283b52c3b6ac7137851/html5/thumbnails/37.jpg)
DefinitionLet X be a smooth algebraic variety. Let Z ⊂ X be a smoothsubvariety and R ⊂ T ∗X be any subvariety. We define therestriction R|Z ⊂ T ∗Z of R to Z by
R|Z := q(p−1X (Z ) ∩ R),
where q : p−1X (Z )→ T ∗Z is the projection.
T ∗X ⊃ p−1Y (Z ) � T ∗Z
Lemma
Let X be a smooth algebraic variety. Let Z ⊂ X be a smoothsubvariety. Let R ⊂ T ∗X be a (weakly) coisotropic variety.Then, under some transversality assumption, R|Z ⊂ T ∗Z is a(weakly) coisotropic variety.
![Page 38: Multiplicity One Theorems](https://reader030.vdocuments.site/reader030/viewer/2022012616/619e7283b52c3b6ac7137851/html5/thumbnails/38.jpg)
Harish-Chandra descent and homogeneity
Notation
S := {(A, v , φ) ∈ Xn|An = 0 and φ(Aiv) = 0∀0 ≤ i ≤ n}.
By Harish-Chandra descent we can assume that anyξ ∈ S∗(X )G̃,χ is supported in S.
Notation
S′ := {(A, v , φ) ∈ S|An−1v = (A∗)n−1φ = 0}.
By the homogeneity theorem, the stratification method andFrobenius descent we get that any ξ ∈ S∗(X )G̃,χ is supported inS′.
D. Gourevitch Multiplicity One Theorems
![Page 39: Multiplicity One Theorems](https://reader030.vdocuments.site/reader030/viewer/2022012616/619e7283b52c3b6ac7137851/html5/thumbnails/39.jpg)
Harish-Chandra descent and homogeneity
Notation
S := {(A, v , φ) ∈ Xn|An = 0 and φ(Aiv) = 0∀0 ≤ i ≤ n}.
By Harish-Chandra descent we can assume that anyξ ∈ S∗(X )G̃,χ is supported in S.
Notation
S′ := {(A, v , φ) ∈ S|An−1v = (A∗)n−1φ = 0}.
By the homogeneity theorem, the stratification method andFrobenius descent we get that any ξ ∈ S∗(X )G̃,χ is supported inS′.
D. Gourevitch Multiplicity One Theorems
![Page 40: Multiplicity One Theorems](https://reader030.vdocuments.site/reader030/viewer/2022012616/619e7283b52c3b6ac7137851/html5/thumbnails/40.jpg)
Harish-Chandra descent and homogeneity
Notation
S := {(A, v , φ) ∈ Xn|An = 0 and φ(Aiv) = 0∀0 ≤ i ≤ n}.
By Harish-Chandra descent we can assume that anyξ ∈ S∗(X )G̃,χ is supported in S.
Notation
S′ := {(A, v , φ) ∈ S|An−1v = (A∗)n−1φ = 0}.
By the homogeneity theorem, the stratification method andFrobenius descent we get that any ξ ∈ S∗(X )G̃,χ is supported inS′.
D. Gourevitch Multiplicity One Theorems
![Page 41: Multiplicity One Theorems](https://reader030.vdocuments.site/reader030/viewer/2022012616/619e7283b52c3b6ac7137851/html5/thumbnails/41.jpg)
Harish-Chandra descent and homogeneity
Notation
S := {(A, v , φ) ∈ Xn|An = 0 and φ(Aiv) = 0∀0 ≤ i ≤ n}.
By Harish-Chandra descent we can assume that anyξ ∈ S∗(X )G̃,χ is supported in S.
Notation
S′ := {(A, v , φ) ∈ S|An−1v = (A∗)n−1φ = 0}.
By the homogeneity theorem, the stratification method andFrobenius descent we get that any ξ ∈ S∗(X )G̃,χ is supported inS′.
D. Gourevitch Multiplicity One Theorems
![Page 42: Multiplicity One Theorems](https://reader030.vdocuments.site/reader030/viewer/2022012616/619e7283b52c3b6ac7137851/html5/thumbnails/42.jpg)
Reduction to the geometric statement
Notation
T ′ = {((A1, v1, φ1), (A2, v2, φ2)) ∈ X × X | ∀i , j ∈ {1,2}(Ai , vj , φj) ∈ S′ and [A1,A2] + v1 ⊗ φ2 − v2 ⊗ φ1 = 0}.
It is enough to show:
Theorem (The geometric statement)
There are no non-empty X × X-weakly coisotropic subvarietiesof T ′.
D. Gourevitch Multiplicity One Theorems
![Page 43: Multiplicity One Theorems](https://reader030.vdocuments.site/reader030/viewer/2022012616/619e7283b52c3b6ac7137851/html5/thumbnails/43.jpg)
Reduction to the geometric statement
Notation
T ′ = {((A1, v1, φ1), (A2, v2, φ2)) ∈ X × X | ∀i , j ∈ {1,2}(Ai , vj , φj) ∈ S′ and [A1,A2] + v1 ⊗ φ2 − v2 ⊗ φ1 = 0}.
It is enough to show:
Theorem (The geometric statement)
There are no non-empty X × X-weakly coisotropic subvarietiesof T ′.
D. Gourevitch Multiplicity One Theorems
![Page 44: Multiplicity One Theorems](https://reader030.vdocuments.site/reader030/viewer/2022012616/619e7283b52c3b6ac7137851/html5/thumbnails/44.jpg)
Reduction to the Key Lemma
Notation
T ′′ := {((A1, v1, φ1), (A2, v2, φ2)) ∈ T ′|An−11 = 0}.
It is easy to see that there are no non-empty X × X -weaklycoisotropic subvarieties of T ′′.
NotationLet A ∈ sl(V ) be a nilpotent Jordan block. DenoteRA := (T ′ − T ′′)|{A}×V×V∗ .
It is enough to show:
Lemma (Key Lemma)There are no non-empty V × V ∗ × V × V ∗-weakly coisotropicsubvarieties of RA.
D. Gourevitch Multiplicity One Theorems
![Page 45: Multiplicity One Theorems](https://reader030.vdocuments.site/reader030/viewer/2022012616/619e7283b52c3b6ac7137851/html5/thumbnails/45.jpg)
Reduction to the Key Lemma
Notation
T ′′ := {((A1, v1, φ1), (A2, v2, φ2)) ∈ T ′|An−11 = 0}.
It is easy to see that there are no non-empty X × X -weaklycoisotropic subvarieties of T ′′.
NotationLet A ∈ sl(V ) be a nilpotent Jordan block. DenoteRA := (T ′ − T ′′)|{A}×V×V∗ .
It is enough to show:
Lemma (Key Lemma)There are no non-empty V × V ∗ × V × V ∗-weakly coisotropicsubvarieties of RA.
D. Gourevitch Multiplicity One Theorems
![Page 46: Multiplicity One Theorems](https://reader030.vdocuments.site/reader030/viewer/2022012616/619e7283b52c3b6ac7137851/html5/thumbnails/46.jpg)
Reduction to the Key Lemma
Notation
T ′′ := {((A1, v1, φ1), (A2, v2, φ2)) ∈ T ′|An−11 = 0}.
It is easy to see that there are no non-empty X × X -weaklycoisotropic subvarieties of T ′′.
NotationLet A ∈ sl(V ) be a nilpotent Jordan block. DenoteRA := (T ′ − T ′′)|{A}×V×V∗ .
It is enough to show:
Lemma (Key Lemma)There are no non-empty V × V ∗ × V × V ∗-weakly coisotropicsubvarieties of RA.
D. Gourevitch Multiplicity One Theorems
![Page 47: Multiplicity One Theorems](https://reader030.vdocuments.site/reader030/viewer/2022012616/619e7283b52c3b6ac7137851/html5/thumbnails/47.jpg)
Reduction to the Key Lemma
Notation
T ′′ := {((A1, v1, φ1), (A2, v2, φ2)) ∈ T ′|An−11 = 0}.
It is easy to see that there are no non-empty X × X -weaklycoisotropic subvarieties of T ′′.
NotationLet A ∈ sl(V ) be a nilpotent Jordan block. DenoteRA := (T ′ − T ′′)|{A}×V×V∗ .
It is enough to show:
Lemma (Key Lemma)There are no non-empty V × V ∗ × V × V ∗-weakly coisotropicsubvarieties of RA.
D. Gourevitch Multiplicity One Theorems
![Page 48: Multiplicity One Theorems](https://reader030.vdocuments.site/reader030/viewer/2022012616/619e7283b52c3b6ac7137851/html5/thumbnails/48.jpg)
Proof of the Key Lemma
Notation
QA := S′ ∩ ({A} × V × V ∗) =n−1⋃i=1
(KerAi)× (Ker(A∗)n−i)
It is easy to see that RA ⊂ QA ×QA and QA ×QA =⋃n−1
i,j=1 Lij ,where
Lij := (KerAi)× (Ker(A∗)n−i)× (KerAj)× (Ker(A∗)n−j).
It is easy to see that any weakly coisotropic subvariety ofQA ×QA is contained in
⋃n−1i=1 Lii . Hence it is enough to show
that for any 0 < i < n, we have dim RA ∩ Lii < 2n. Let f ∈ O(Lii)be the polynomial defined by
f (v1, φ1, v2, φ2) := (v1)i(φ2)i+1 − (v2)i(φ1)i+1.
It is enough to show that f (RA ∩ Lii) = {0}.
D. Gourevitch Multiplicity One Theorems
![Page 49: Multiplicity One Theorems](https://reader030.vdocuments.site/reader030/viewer/2022012616/619e7283b52c3b6ac7137851/html5/thumbnails/49.jpg)
Proof of the Key Lemma
Notation
QA := S′ ∩ ({A} × V × V ∗) =n−1⋃i=1
(KerAi)× (Ker(A∗)n−i)
It is easy to see that RA ⊂ QA ×QA
and QA ×QA =⋃n−1
i,j=1 Lij ,where
Lij := (KerAi)× (Ker(A∗)n−i)× (KerAj)× (Ker(A∗)n−j).
It is easy to see that any weakly coisotropic subvariety ofQA ×QA is contained in
⋃n−1i=1 Lii . Hence it is enough to show
that for any 0 < i < n, we have dim RA ∩ Lii < 2n. Let f ∈ O(Lii)be the polynomial defined by
f (v1, φ1, v2, φ2) := (v1)i(φ2)i+1 − (v2)i(φ1)i+1.
It is enough to show that f (RA ∩ Lii) = {0}.
D. Gourevitch Multiplicity One Theorems
![Page 50: Multiplicity One Theorems](https://reader030.vdocuments.site/reader030/viewer/2022012616/619e7283b52c3b6ac7137851/html5/thumbnails/50.jpg)
Proof of the Key Lemma
Notation
QA := S′ ∩ ({A} × V × V ∗) =n−1⋃i=1
(KerAi)× (Ker(A∗)n−i)
It is easy to see that RA ⊂ QA ×QA and QA ×QA =⋃n−1
i,j=1 Lij ,where
Lij := (KerAi)× (Ker(A∗)n−i)× (KerAj)× (Ker(A∗)n−j).
It is easy to see that any weakly coisotropic subvariety ofQA ×QA is contained in
⋃n−1i=1 Lii . Hence it is enough to show
that for any 0 < i < n, we have dim RA ∩ Lii < 2n. Let f ∈ O(Lii)be the polynomial defined by
f (v1, φ1, v2, φ2) := (v1)i(φ2)i+1 − (v2)i(φ1)i+1.
It is enough to show that f (RA ∩ Lii) = {0}.
D. Gourevitch Multiplicity One Theorems
![Page 51: Multiplicity One Theorems](https://reader030.vdocuments.site/reader030/viewer/2022012616/619e7283b52c3b6ac7137851/html5/thumbnails/51.jpg)
Proof of the Key Lemma
Notation
QA := S′ ∩ ({A} × V × V ∗) =n−1⋃i=1
(KerAi)× (Ker(A∗)n−i)
It is easy to see that RA ⊂ QA ×QA and QA ×QA =⋃n−1
i,j=1 Lij ,where
Lij := (KerAi)× (Ker(A∗)n−i)× (KerAj)× (Ker(A∗)n−j).
It is easy to see that any weakly coisotropic subvariety ofQA ×QA is contained in
⋃n−1i=1 Lii .
Hence it is enough to showthat for any 0 < i < n, we have dim RA ∩ Lii < 2n. Let f ∈ O(Lii)be the polynomial defined by
f (v1, φ1, v2, φ2) := (v1)i(φ2)i+1 − (v2)i(φ1)i+1.
It is enough to show that f (RA ∩ Lii) = {0}.
D. Gourevitch Multiplicity One Theorems
![Page 52: Multiplicity One Theorems](https://reader030.vdocuments.site/reader030/viewer/2022012616/619e7283b52c3b6ac7137851/html5/thumbnails/52.jpg)
Proof of the Key Lemma
Notation
QA := S′ ∩ ({A} × V × V ∗) =n−1⋃i=1
(KerAi)× (Ker(A∗)n−i)
It is easy to see that RA ⊂ QA ×QA and QA ×QA =⋃n−1
i,j=1 Lij ,where
Lij := (KerAi)× (Ker(A∗)n−i)× (KerAj)× (Ker(A∗)n−j).
It is easy to see that any weakly coisotropic subvariety ofQA ×QA is contained in
⋃n−1i=1 Lii . Hence it is enough to show
that for any 0 < i < n, we have dim RA ∩ Lii < 2n.
Let f ∈ O(Lii)be the polynomial defined by
f (v1, φ1, v2, φ2) := (v1)i(φ2)i+1 − (v2)i(φ1)i+1.
It is enough to show that f (RA ∩ Lii) = {0}.
D. Gourevitch Multiplicity One Theorems
![Page 53: Multiplicity One Theorems](https://reader030.vdocuments.site/reader030/viewer/2022012616/619e7283b52c3b6ac7137851/html5/thumbnails/53.jpg)
Proof of the Key Lemma
Notation
QA := S′ ∩ ({A} × V × V ∗) =n−1⋃i=1
(KerAi)× (Ker(A∗)n−i)
It is easy to see that RA ⊂ QA ×QA and QA ×QA =⋃n−1
i,j=1 Lij ,where
Lij := (KerAi)× (Ker(A∗)n−i)× (KerAj)× (Ker(A∗)n−j).
It is easy to see that any weakly coisotropic subvariety ofQA ×QA is contained in
⋃n−1i=1 Lii . Hence it is enough to show
that for any 0 < i < n, we have dim RA ∩ Lii < 2n. Let f ∈ O(Lii)be the polynomial defined by
f (v1, φ1, v2, φ2) := (v1)i(φ2)i+1 − (v2)i(φ1)i+1.
It is enough to show that f (RA ∩ Lii) = {0}.
D. Gourevitch Multiplicity One Theorems
![Page 54: Multiplicity One Theorems](https://reader030.vdocuments.site/reader030/viewer/2022012616/619e7283b52c3b6ac7137851/html5/thumbnails/54.jpg)
Proof of the Key Lemma
Notation
QA := S′ ∩ ({A} × V × V ∗) =n−1⋃i=1
(KerAi)× (Ker(A∗)n−i)
It is easy to see that RA ⊂ QA ×QA and QA ×QA =⋃n−1
i,j=1 Lij ,where
Lij := (KerAi)× (Ker(A∗)n−i)× (KerAj)× (Ker(A∗)n−j).
It is easy to see that any weakly coisotropic subvariety ofQA ×QA is contained in
⋃n−1i=1 Lii . Hence it is enough to show
that for any 0 < i < n, we have dim RA ∩ Lii < 2n. Let f ∈ O(Lii)be the polynomial defined by
f (v1, φ1, v2, φ2) := (v1)i(φ2)i+1 − (v2)i(φ1)i+1.
It is enough to show that f (RA ∩ Lii) = {0}.D. Gourevitch Multiplicity One Theorems
![Page 55: Multiplicity One Theorems](https://reader030.vdocuments.site/reader030/viewer/2022012616/619e7283b52c3b6ac7137851/html5/thumbnails/55.jpg)
Proof of the Key Lemma
Let (v1, φ1, v2, φ2) ∈ Lii . Let M := v1 ⊗ φ2 − v2 ⊗ φ1.
Clearly, Mis of the form
M =
(0i×i ∗
0(n−i)×i 0(n−i)×(n−i)
).
We know that there exists a nilpotent B satisfying [A,B] = M.Hence this B is upper nilpotent, which implies Mi,i+1 = 0 andhence f (v1, φ1, v2, φ2) = 0.
D. Gourevitch Multiplicity One Theorems
![Page 56: Multiplicity One Theorems](https://reader030.vdocuments.site/reader030/viewer/2022012616/619e7283b52c3b6ac7137851/html5/thumbnails/56.jpg)
Proof of the Key Lemma
Let (v1, φ1, v2, φ2) ∈ Lii . Let M := v1 ⊗ φ2 − v2 ⊗ φ1. Clearly, Mis of the form
M =
(0i×i ∗
0(n−i)×i 0(n−i)×(n−i)
).
We know that there exists a nilpotent B satisfying [A,B] = M.Hence this B is upper nilpotent, which implies Mi,i+1 = 0 andhence f (v1, φ1, v2, φ2) = 0.
D. Gourevitch Multiplicity One Theorems
![Page 57: Multiplicity One Theorems](https://reader030.vdocuments.site/reader030/viewer/2022012616/619e7283b52c3b6ac7137851/html5/thumbnails/57.jpg)
Proof of the Key Lemma
Let (v1, φ1, v2, φ2) ∈ Lii . Let M := v1 ⊗ φ2 − v2 ⊗ φ1. Clearly, Mis of the form
M =
(0i×i ∗
0(n−i)×i 0(n−i)×(n−i)
).
We know that there exists a nilpotent B satisfying [A,B] = M.
Hence this B is upper nilpotent, which implies Mi,i+1 = 0 andhence f (v1, φ1, v2, φ2) = 0.
D. Gourevitch Multiplicity One Theorems
![Page 58: Multiplicity One Theorems](https://reader030.vdocuments.site/reader030/viewer/2022012616/619e7283b52c3b6ac7137851/html5/thumbnails/58.jpg)
Proof of the Key Lemma
Let (v1, φ1, v2, φ2) ∈ Lii . Let M := v1 ⊗ φ2 − v2 ⊗ φ1. Clearly, Mis of the form
M =
(0i×i ∗
0(n−i)×i 0(n−i)×(n−i)
).
We know that there exists a nilpotent B satisfying [A,B] = M.Hence this B is upper nilpotent, which implies Mi,i+1 = 0
andhence f (v1, φ1, v2, φ2) = 0.
D. Gourevitch Multiplicity One Theorems
![Page 59: Multiplicity One Theorems](https://reader030.vdocuments.site/reader030/viewer/2022012616/619e7283b52c3b6ac7137851/html5/thumbnails/59.jpg)
Proof of the Key Lemma
Let (v1, φ1, v2, φ2) ∈ Lii . Let M := v1 ⊗ φ2 − v2 ⊗ φ1. Clearly, Mis of the form
M =
(0i×i ∗
0(n−i)×i 0(n−i)×(n−i)
).
We know that there exists a nilpotent B satisfying [A,B] = M.Hence this B is upper nilpotent, which implies Mi,i+1 = 0 andhence f (v1, φ1, v2, φ2) = 0.
D. Gourevitch Multiplicity One Theorems
![Page 60: Multiplicity One Theorems](https://reader030.vdocuments.site/reader030/viewer/2022012616/619e7283b52c3b6ac7137851/html5/thumbnails/60.jpg)
Summary
Flowchart
sl(V )× V × V ∗
D. Gourevitch Multiplicity One Theorems
![Page 61: Multiplicity One Theorems](https://reader030.vdocuments.site/reader030/viewer/2022012616/619e7283b52c3b6ac7137851/html5/thumbnails/61.jpg)
Summary
Flowchart
sl(V )× V × V ∗ H.Ch.
descent// S
D. Gourevitch Multiplicity One Theorems
![Page 62: Multiplicity One Theorems](https://reader030.vdocuments.site/reader030/viewer/2022012616/619e7283b52c3b6ac7137851/html5/thumbnails/62.jpg)
Summary
Flowchart
sl(V )× V × V ∗ H.Ch.
descent// S
homogeneity theorem
Fourier transform and // S′
D. Gourevitch Multiplicity One Theorems
![Page 63: Multiplicity One Theorems](https://reader030.vdocuments.site/reader030/viewer/2022012616/619e7283b52c3b6ac7137851/html5/thumbnails/63.jpg)
Summary
Flowchart
sl(V )× V × V ∗ H.Ch.
descent// S
homogeneity theorem
Fourier transform and // S′integrability theorem
Fourier transform and // T ′
D. Gourevitch Multiplicity One Theorems
![Page 64: Multiplicity One Theorems](https://reader030.vdocuments.site/reader030/viewer/2022012616/619e7283b52c3b6ac7137851/html5/thumbnails/64.jpg)
Summary
Flowchart
sl(V )× V × V ∗ H.Ch.
descent// S
homogeneity theorem
Fourier transform and // S′integrability theorem
Fourier transform and // T ′
��T ′ − T ′′
D. Gourevitch Multiplicity One Theorems
![Page 65: Multiplicity One Theorems](https://reader030.vdocuments.site/reader030/viewer/2022012616/619e7283b52c3b6ac7137851/html5/thumbnails/65.jpg)
Summary
Flowchart
sl(V )× V × V ∗ H.Ch.
descent// S
homogeneity theorem
Fourier transform and // S′integrability theorem
Fourier transform and // T ′
��RA T ′ − T ′′
restrictionoo
D. Gourevitch Multiplicity One Theorems
![Page 66: Multiplicity One Theorems](https://reader030.vdocuments.site/reader030/viewer/2022012616/619e7283b52c3b6ac7137851/html5/thumbnails/66.jpg)
Summary
Flowchart
sl(V )× V × V ∗ H.Ch.
descent// S
homogeneity theorem
Fourier transform and // S′integrability theorem
Fourier transform and // T ′
��Lii ∩RA RA
RA⊂⋃
Lijoo T ′ − T ′′restrictionoo
D. Gourevitch Multiplicity One Theorems
![Page 67: Multiplicity One Theorems](https://reader030.vdocuments.site/reader030/viewer/2022012616/619e7283b52c3b6ac7137851/html5/thumbnails/67.jpg)
Summary
Flowchart
sl(V )× V × V ∗ H.Ch.
descent// S
homogeneity theorem
Fourier transform and // S′integrability theorem
Fourier transform and // T ′
��∅ Lii ∩RA
f (RA∩Lii )=0oo RARA⊂
⋃Lijoo T ′ − T ′′
restrictionoo
D. Gourevitch Multiplicity One Theorems