Multiplicity One Theorem for GL(N) and Other Gelfand Pairs

Multiplicity one theorem for GL(n) and other Gelfand pairs

A. Aizenbud

Massachusetts Institute of Technology

http://math.mit.edu/~aizenr

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs L2(S1) = L span{einx } geinx = χ(g)einx

Example (Spherical Harmonics) 2 2 L m L (S ) = m H m m H = spani {yi } are irreducible representations of O3

Example Let X be a ﬁnite set. Let the symmetric group Perm(X) act on X. Consider the space F(X) of complex valued functions on X as a representation of Perm(X). Then it decomposes to direct sum of distinct irreducible representations.

Examples

Example (Fourier Series)

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs geinx = χ(g)einx

Example (Spherical Harmonics) 2 2 L m L (S ) = m H m m H = spani {yi } are irreducible representations of O3

Examples

Example (Fourier Series) L2(S1) = L span{einx }

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Example (Spherical Harmonics) 2 2 L m L (S ) = m H m m H = spani {yi } are irreducible representations of O3

Examples

Example (Fourier Series) L2(S1) = L span{einx } geinx = χ(g)einx

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs 2 2 L m L (S ) = m H m m H = spani {yi } are irreducible representations of O3

Examples

Example (Fourier Series) L2(S1) = L span{einx } geinx = χ(g)einx

Example (Spherical Harmonics)

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs m m H = spani {yi } are irreducible representations of O3

Examples

Example (Fourier Series) L2(S1) = L span{einx } geinx = χ(g)einx

Example (Spherical Harmonics) 2 2 L m L (S ) = m H

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Example Let X be a ﬁnite set. Let the symmetric group Perm(X) act on X. Consider the space F(X) of complex valued functions on X as a representation of Perm(X). Then it decomposes to direct sum of distinct irreducible representations.

Examples

Example (Fourier Series) L2(S1) = L span{einx } geinx = χ(g)einx

Example (Spherical Harmonics) 2 2 L m L (S ) = m H m m H = spani {yi } are irreducible representations of O3

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Examples

Example (Fourier Series) L2(S1) = L span{einx } geinx = χ(g)einx

Example (Spherical Harmonics) 2 2 L m L (S ) = m H m m H = spani {yi } are irreducible representations of O3

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Examples

Example (Fourier Series) L2(S1) = L span{einx } geinx = χ(g)einx

Example (Spherical Harmonics) 2 2 L m L (S ) = m H m m H = spani {yi } are irreducible representations of O3

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs L2(G/H) decomposes to direct sum of distinct irreducible representations of G. for any irreducible representation ρ of G, dimρH ≤ 1. for any irreducible representation ρ of G, dimHomH (ρ, C) ≤ 1. the algebra of bi-H-invariant functions on G, C(H \G/H), is commutative w.r.t. convolution.

Gelfand Pairs

Deﬁnition A pair of compact topological groups (G ⊃ H) is called a Gelfand pair if the following equivalent conditions hold: for any irreducible representation ρ of G, dimρH ≤ 1. for any irreducible representation ρ of G, dimHomH (ρ, C) ≤ 1. the algebra of bi-H-invariant functions on G, C(H \G/H), is commutative w.r.t. convolution.

Gelfand Pairs

Deﬁnition A pair of compact topological groups (G ⊃ H) is called a Gelfand pair if the following equivalent conditions hold: L2(G/H) decomposes to direct sum of distinct irreducible representations of G. for any irreducible representation ρ of G, dimρH ≤ 1. for any irreducible representation ρ of G, dimHomH (ρ, C) ≤ 1. the algebra of bi-H-invariant functions on G, C(H \G/H), is commutative w.r.t. convolution. Conclusion Let H ⊂ G be a pair of groups. One can consider harmonic analysis over G/H as a generalization of representation theory.

Example Schur’s lemma is equivalent to the Gelfand property of (G × G, ∆G):

∀π ⊗ ρ ∈ irr(G × G): dim(π ⊗ ρ)∆G ≤ 1 m ∗ dim HomG(π , ρ) ≤ 1

Philosophical motivation

Observation Representation theory of G m Harmonic analysis on G w.r.t. the two sided action of G × G Example Schur’s lemma is equivalent to the Gelfand property of (G × G, ∆G):

∀π ⊗ ρ ∈ irr(G × G): dim(π ⊗ ρ)∆G ≤ 1 m ∗ dim HomG(π , ρ) ≤ 1

Philosophical motivation

Observation Representation theory of G m Harmonic analysis on G w.r.t. the two sided action of G × G

Conclusion Let H ⊂ G be a pair of groups. One can consider harmonic analysis over G/H as a generalization of representation theory. ∀π ⊗ ρ ∈ irr(G × G): dim(π ⊗ ρ)∆G ≤ 1 m ∗ dim HomG(π , ρ) ≤ 1

Philosophical motivation

Observation Representation theory of G m Harmonic analysis on G w.r.t. the two sided action of G × G

Conclusion Let H ⊂ G be a pair of groups. One can consider harmonic analysis over G/H as a generalization of representation theory.

Example Schur’s lemma is equivalent to the Gelfand property of (G × G, ∆G): m ∗ dim HomG(π , ρ) ≤ 1

Philosophical motivation

Observation Representation theory of G m Harmonic analysis on G w.r.t. the two sided action of G × G

Conclusion Let H ⊂ G be a pair of groups. One can consider harmonic analysis over G/H as a generalization of representation theory.

Example Schur’s lemma is equivalent to the Gelfand property of (G × G, ∆G):

∀π ⊗ ρ ∈ irr(G × G): dim(π ⊗ ρ)∆G ≤ 1 Philosophical motivation

Observation Representation theory of G m Harmonic analysis on G w.r.t. the two sided action of G × G

Conclusion Let H ⊂ G be a pair of groups. One can consider harmonic analysis over G/H as a generalization of representation theory.

Example Schur’s lemma is equivalent to the Gelfand property of (G × G, ∆G):

∀π ⊗ ρ ∈ irr(G × G): dim(π ⊗ ρ)∆G ≤ 1 m ∗ dim HomG(π , ρ) ≤ 1 the pair (G × H ⊃ ∆H) is a Gelfand pair for any irreducible representations ρ of G and τ of H,

dimHomH (ρ|H , τ) ≤ 1.

the algebra of Ad(H)-invariant functions on G, C(G//H) := C(G/Ad(H)), is commutative w.r.t. convolution.

Strong Gelfand Pairs

Deﬁnition A pair of compact topological groups (G ⊃ H) is called a strong Gelfand pair if one of the following equivalent conditions is satisﬁed:

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs for any irreducible representations ρ of G and τ of H,

dimHomH (ρ|H , τ) ≤ 1.

the algebra of Ad(H)-invariant functions on G, C(G//H) := C(G/Ad(H)), is commutative w.r.t. convolution.

Strong Gelfand Pairs

Deﬁnition A pair of compact topological groups (G ⊃ H) is called a strong Gelfand pair if one of the following equivalent conditions is satisﬁed: the pair (G × H ⊃ ∆H) is a Gelfand pair

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs the algebra of Ad(H)-invariant functions on G, C(G//H) := C(G/Ad(H)), is commutative w.r.t. convolution.

Strong Gelfand Pairs

Deﬁnition A pair of compact topological groups (G ⊃ H) is called a strong Gelfand pair if one of the following equivalent conditions is satisﬁed: the pair (G × H ⊃ ∆H) is a Gelfand pair for any irreducible representations ρ of G and τ of H,

dimHomH (ρ|H , τ) ≤ 1.

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Strong Gelfand Pairs

dimHomH (ρ|H , τ) ≤ 1.

the algebra of Ad(H)-invariant functions on G, C(G//H) := C(G/Ad(H)), is commutative w.r.t. convolution.

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Proposition (Gelfand) Let σ be an involutive anti-automorphism of G (i.e. 2 σ(g1g2) = σ(g2)σ(g1)) and σ = Id and assume σ(H) = H. Suppose that σ(f ) = f for all Ad(H)-invariant functions f ∈ C(G//H). Then (G, H) is a strong Gelfand pair.

Gelfand trick

Proposition (Gelfand) Let σ be an involutive anti-automorphism of G (i.e. 2 σ(g1g2) = σ(g2)σ(g1) and σ = Id) and assume σ(H) = H. Suppose that σ(f ) = f for all bi H-invariant functions f ∈ C(H \G/H). Then (G, H) is a Gelfand pair. Gelfand trick

Proposition (Gelfand) Let σ be an involutive anti-automorphism of G (i.e. 2 σ(g1g2) = σ(g2)σ(g1)) and σ = Id and assume σ(H) = H. Suppose that σ(f ) = f for all Ad(H)-invariant functions f ∈ C(G//H). Then (G, H) is a strong Gelfand pair. Sum up

"Analysis": Algebra: Rep. theory: ∃ anti-involution σ ks +3 C(H\G/H) is ks ∀ρ dim ρH ≤ 1 s.t. f = σ(f ) commutative ∀f ∈ C(H \G/H) KS Geometry: ∃ anti-involution σ that preserves H double cosets

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Definition A local field is a locally compact non-discrete topological field. There are 2 types of local fields of characteristic zero: Archimedean: R and C non-Archimedean: Qp and their finite extensions

Deﬁnition

A linear algebraic group is a subgroup of GLn deﬁned by polynomial equations.

Non compact setting

Setting In the non compact case we will consider complex smooth admissible representations of algebraic reductive groups over local ﬁelds.

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Deﬁnition

A linear algebraic group is a subgroup of GLn deﬁned by polynomial equations.

Non compact setting

Setting In the non compact case we will consider complex smooth admissible representations of algebraic reductive groups over local ﬁelds.

Definition A local field is a locally compact non-discrete topological field. There are 2 types of local fields of characteristic zero: Archimedean: R and C non-Archimedean: Qp and their finite extensions

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Non compact setting

Setting In the non compact case we will consider complex smooth admissible representations of algebraic reductive groups over local ﬁelds.

Deﬁnition

A linear algebraic group is a subgroup of GLn deﬁned by polynomial equations.

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Fact Any algebraic representation of a reductive group decomposes to a direct sum of irreducible representations.

Fact Reductive groups are unimodular.

Reductive groups

Examples

GLn, On, Un, Sp2n,..., semisimple groups,

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Fact Reductive groups are unimodular.

Reductive groups

Examples

GLn, On, Un, Sp2n,..., semisimple groups,

Fact Any algebraic representation of a reductive group decomposes to a direct sum of irreducible representations.

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Reductive groups

Examples

GLn, On, Un, Sp2n,..., semisimple groups,

Fact Any algebraic representation of a reductive group decomposes to a direct sum of irreducible representations.

Fact Reductive groups are unimodular.

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Deﬁnition Over non-Archimedean F, by smooth representation V we mean a complex linear representation V such that for any v ∈ V there exists an open compact subgroup K < G such that Kv = v.

Smooth representations

Deﬁnition Over Archimedean F, by smooth representation V we mean a complex Fréchet representation V such that for any v ∈ V the map G → V deﬁned by v is smooth.

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Smooth representations

Deﬁnition Over Archimedean F, by smooth representation V we mean a complex Fréchet representation V such that for any v ∈ V the map G → V deﬁned by v is smooth.

Deﬁnition Over non-Archimedean F, by smooth representation V we mean a complex linear representation V such that for any v ∈ V there exists an open compact subgroup K < G such that Kv = v.

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Deﬁnition An `-space is a Hausdorff locally compact totally disconnected topological space. For an `-space X we denote by S(X) the space of compactly supported locally constant functions on X. We let S∗(X) := S(X)∗ be the space of distributions on X.

Distributions

Notation ∞ Let M be a smooth manifold. We denote by Cc (M) the space of smooth compactly supported functions on M. We will ∞ ∗ consider the space (Cc (M)) of distributions on M. Sometimes we will also consider the space S∗(M) of Schwartz distributions on M.

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Distributions

Deﬁnition An `-space is a Hausdorff locally compact totally disconnected topological space. For an `-space X we denote by S(X) the space of compactly supported locally constant functions on X. We let S∗(X) := S(X)∗ be the space of distributions on X.

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Gelfand Pairs

Deﬁnition A pair of groups (G ⊃ H) is called a Gelfand pair if for any irreducible admissible representation ρ of G

dimHomH (ρ, C) · dimHomH (ρ,e C) ≤ 1 Usually, this implies that

dimHomH (ρ, C) ≤ 1.

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Gelfand-Kazhdan distributional criterion

Theorem (Gelfand-Kazhdan,...) Let σ be an involutive anti-automorphism of G and assume σ(H) = H. Suppose that σ(ξ) = ξ for all bi H-invariant distributions ξ on G. Then (G, H) is a Gelfand pair.

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Proposition The pair (G, H) is a strong Gelfand pair if and only if the pair (G × H, ∆H) is a Gelfand pair.

Strong Gelfand Pairs

Deﬁnition A pair of groups (G, H) is called a strong Gelfand pair if for any irreducible admissible representations ρ of G and τ of H

dimHomH (ρ, τ) · dimHomH (ρ,e τe) ≤ 1

Usually, this implies that dimHomH (ρ, τ) ≤ 1. Corollary Let σ be an involutive anti-automorphism of G s.t. σ(H) = H. Suppose σ(ξ) = ξ for all distributions ξ on G invariant with respect to conjugation by H. Then (G, H) is a strong Gelfand pair.

Strong Gelfand Pairs

Deﬁnition A pair of groups (G, H) is called a strong Gelfand pair if for any irreducible admissible representations ρ of G and τ of H

dimHomH (ρ, τ) · dimHomH (ρ,e τe) ≤ 1

Usually, this implies that dimHomH (ρ, τ) ≤ 1.

Proposition The pair (G, H) is a strong Gelfand pair if and only if the pair (G × H, ∆H) is a Gelfand pair. Strong Gelfand Pairs

Deﬁnition A pair of groups (G, H) is called a strong Gelfand pair if for any irreducible admissible representations ρ of G and τ of H

dimHomH (ρ, τ) · dimHomH (ρ,e τe) ≤ 1

Usually, this implies that dimHomH (ρ, τ) ≤ 1.

Proposition The pair (G, H) is a strong Gelfand pair if and only if the pair (G × H, ∆H) is a Gelfand pair.

Corollary Let σ be an involutive anti-automorphism of G s.t. σ(H) = H. Suppose σ(ξ) = ξ for all distributions ξ on G invariant with respect to conjugation by H. Then (G, H) is a strong Gelfand pair. Rep. theory: ow Analysis: dim HomH (ρ, C) ≤ 1 Algebra ∃ σ s.t. ξ = σ(ξ) ∀ρ ∀ξ ∈ S∗(G)H×H ss KS sssss sssss p−adic u} ssss Geometry: Geometry: ∃ σ that preserves Non − compact case ks ∃ σ that preserves closed H double cosets H double cosets

Algebra: "Analysis": Rep. theory: ks +3 C(H \G/H) ks ∃ σ s.t. f = σ(f ) ∀ρ dim ρH ≤ 1 is commutative ∀f ∈ C(H \G/H) KS

Geometry: Compact case ∃ σ that preserves H double cosets Algebra: "Analysis": Rep. theory: ks +3 C(H \G/H) ks ∃ σ s.t. f = σ(f ) ∀ρ dim ρH ≤ 1 is commutative ∀f ∈ C(H \G/H) KS

Geometry: Compact case ∃ σ that preserves H double cosets

Rep. theory: ow Analysis: dim HomH (ρ, C) ≤ 1 Algebra ∃ σ s.t. ξ = σ(ξ) ∀ρ ∀ξ ∈ S∗(G)H×H ss KS sssss sssss p−adic u} ssss Geometry: Geometry: ∃ σ that preserves Non − compact case ks ∃ σ that preserves closed H double cosets H double cosets It has the following corollary in representation theory. Theorem (A.-Gourevitch-Rallis-Schiffmann-Sun-Zhu)

Let 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.

Multiplicity one theorem for GLn

Let F be a local ﬁeld of characteristic zero. Theorem (A.-Gourevitch-Rallis-Schiffmann-Sun-Zhu)

Every GLn(F)-invariant distribution on GLn+1(F) is transposition invariant.

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Similar theorems hold for orthogonal and unitary groups.

Multiplicity one theorem for GLn

Let F be a local ﬁeld of characteristic zero. Theorem (A.-Gourevitch-Rallis-Schiffmann-Sun-Zhu)

Every GLn(F)-invariant distribution on GLn+1(F) is transposition invariant.

It has the following corollary in representation theory. Theorem (A.-Gourevitch-Rallis-Schiffmann-Sun-Zhu)

Let 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.

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Multiplicity one theorem for GLn

Let F be a local ﬁeld of characteristic zero. Theorem (A.-Gourevitch-Rallis-Schiffmann-Sun-Zhu)

Every GLn(F)-invariant distribution on GLn+1(F) is transposition invariant.

It has the following corollary in representation theory. Theorem (A.-Gourevitch-Rallis-Schiffmann-Sun-Zhu)

Let 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.

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Ge := GLn(F) o {1, σ}

Deﬁne a character χ of Ge by χ(GLn(F)) = {1}, χ(Ge − GLn(F)) = {−1}.

Equivalent formulation: Theorem

∗ Ge,χ S (GLn+1(F)) = 0.

Reformulation

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Equivalent formulation: Theorem

∗ Ge,χ S (GLn+1(F)) = 0.

Reformulation

Ge := GLn(F) o {1, σ}

Deﬁne a character χ of Ge by χ(GLn(F)) = {1}, χ(Ge − GLn(F)) = {−1}.

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Reformulation

Ge := GLn(F) o {1, σ}

Deﬁne a character χ of Ge by χ(GLn(F)) = {1}, χ(Ge − GLn(F)) = {−1}.

Equivalent formulation: Theorem

∗ Ge,χ S (GLn+1(F)) = 0.

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs −1 t t t An×n vn×1 −1 gAg gv A v A φ g g = ∗ −1 and = t φ1×n λ (g ) φ λ φ λ v λ

V := F n X := sl(V ) × V × V ∗ Ge acts on X by g(A, v, φ) = (gAg−1, gv, (g∗)−1φ) σ(A, v, φ) = (At , φt , v t ). Equivalent formulation: Theorem S∗(X)Ge,χ = 0.

Equivalent formulation: Theorem

∗ Ge,χ S (gln+1(F)) = 0. V := F n X := sl(V ) × V × V ∗ Ge acts on X by g(A, v, φ) = (gAg−1, gv, (g∗)−1φ) σ(A, v, φ) = (At , φt , v t ). Equivalent formulation: Theorem S∗(X)Ge,χ = 0.

Equivalent formulation: Theorem

∗ Ge,χ S (gln+1(F)) = 0.

−1 t t t An×n vn×1 −1 gAg gv A v A φ g g = ∗ −1 and = t φ1×n λ (g ) φ λ φ λ v λ Ge acts on X by g(A, v, φ) = (gAg−1, gv, (g∗)−1φ) σ(A, v, φ) = (At , φt , v t ). Equivalent formulation: Theorem S∗(X)Ge,χ = 0.

Equivalent formulation: Theorem

∗ Ge,χ S (gln+1(F)) = 0.

−1 t t t An×n vn×1 −1 gAg gv A v A φ g g = ∗ −1 and = t φ1×n λ (g ) φ λ φ λ v λ

V := F n X := sl(V ) × V × V ∗ Equivalent formulation: Theorem S∗(X)Ge,χ = 0.

Equivalent formulation: Theorem

∗ Ge,χ S (gln+1(F)) = 0.

−1 t t t An×n vn×1 −1 gAg gv A v A φ g g = ∗ −1 and = t φ1×n λ (g ) φ λ φ λ v λ

V := F n X := sl(V ) × V × V ∗ Ge acts on X by g(A, v, φ) = (gAg−1, gv, (g∗)−1φ) σ(A, v, φ) = (At , φt , v t ). Equivalent formulation: Theorem

∗ Ge,χ S (gln+1(F)) = 0.

−1 t t t An×n vn×1 −1 gAg gv A v A φ g g = ∗ −1 and = t φ1×n λ (g ) φ λ φ λ v λ

V := F n X := sl(V ) × V × V ∗ Ge acts on X by g(A, v, φ) = (gAg−1, gv, (g∗)−1φ) σ(A, v, φ) = (At , φt , v t ). Equivalent formulation: Theorem S∗(X)Ge,χ = 0. Proposition Let U ⊂ X be an open G-invariant subset and Z := X − U. ∗ G,χ ∗ G,χ Suppose that S (U) = 0 and SX (Z) = 0. Then S∗(X)G,χ = 0.

Proof. ∗ G,χ ∗ G,χ ∗ G,χ 0 → SX (Z ) → S (X) → S (U) . ∗ G,χ ∼ ∗ G,χ For `-spaces, SX (Z) = S (Z) . For smooth manifolds, there is a slightly more complicated statement which takes into account transversal derivatives: ∗ ∗ k X grk (SX (Z)) = S (Z , Sym (CNZ ))

First tool: Stratiﬁcation

Setting A group G acts on a space X, and χ is a character of G. We want to show S∗(X)G,χ = 0.

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Proposition Let U ⊂ X be an open G-invariant subset and Z := X − U. ∗ G,χ ∗ G,χ Suppose that S (U) = 0 and SX (Z) = 0. Then S∗(X)G,χ = 0.

First tool: Stratiﬁcation

Setting A group G acts on a space X, and χ is a character of G. We want to show S∗(X)G,χ = 0.

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Proof. ∗ G,χ ∗ G,χ ∗ G,χ 0 → SX (Z ) → S (X) → S (U) . ∗ G,χ ∼ ∗ G,χ For `-spaces, SX (Z) = S (Z) . For smooth manifolds, there is a slightly more complicated statement which takes into account transversal derivatives: ∗ ∗ k X grk (SX (Z)) = S (Z , Sym (CNZ ))

First tool: Stratiﬁcation

Setting A group G acts on a space X, and χ is a character of G. We want to show S∗(X)G,χ = 0.

Proposition Let U ⊂ X be an open G-invariant subset and Z := X − U. ∗ G,χ ∗ G,χ Suppose that S (U) = 0 and SX (Z) = 0. Then S∗(X)G,χ = 0.

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs ∗ G,χ ∼ ∗ G,χ For `-spaces, SX (Z) = S (Z) . For smooth manifolds, there is a slightly more complicated statement which takes into account transversal derivatives: ∗ ∗ k X grk (SX (Z)) = S (Z , Sym (CNZ ))

First tool: Stratiﬁcation

Setting A group G acts on a space X, and χ is a character of G. We want to show S∗(X)G,χ = 0.

Proposition Let U ⊂ X be an open G-invariant subset and Z := X − U. ∗ G,χ ∗ G,χ Suppose that S (U) = 0 and SX (Z) = 0. Then S∗(X)G,χ = 0.

Proof. ∗ G,χ ∗ G,χ ∗ G,χ 0 → SX (Z ) → S (X) → S (U) .

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs For smooth manifolds, there is a slightly more complicated statement which takes into account transversal derivatives: ∗ ∗ k X grk (SX (Z)) = S (Z , Sym (CNZ ))

First tool: Stratiﬁcation

Setting A group G acts on a space X, and χ is a character of G. We want to show S∗(X)G,χ = 0.

Proposition Let U ⊂ X be an open G-invariant subset and Z := X − U. ∗ G,χ ∗ G,χ Suppose that S (U) = 0 and SX (Z) = 0. Then S∗(X)G,χ = 0.

Proof. ∗ G,χ ∗ G,χ ∗ G,χ 0 → SX (Z ) → S (X) → S (U) . ∗ G,χ ∼ ∗ G,χ For `-spaces, SX (Z) = S (Z) .

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs ∗ ∗ k X grk (SX (Z)) = S (Z , Sym (CNZ ))

First tool: Stratiﬁcation

Setting A group G acts on a space X, and χ is a character of G. We want to show S∗(X)G,χ = 0.

Proposition Let U ⊂ X be an open G-invariant subset and Z := X − U. ∗ G,χ ∗ G,χ Suppose that S (U) = 0 and SX (Z) = 0. Then S∗(X)G,χ = 0.

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs First tool: Stratiﬁcation

Setting A group G acts on a space X, and χ is a character of G. We want to show S∗(X)G,χ = 0.

Proposition Let U ⊂ X be an open G-invariant subset and Z := X − U. ∗ G,χ ∗ G,χ Suppose that S (U) = 0 and SX (Z) = 0. Then S∗(X)G,χ = 0.

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

∗ G,χ ∼ ∗ StabG(z),χ S (X) = S (Xz ) .

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Generalized Harish-Chandra descent

Theorem (A.-Gourevitch) Let a reductive group G act on a smooth afﬁne algebraic variety X. Let χ be a character of G. Suppose that for any a ∈ X s.t. the orbit Ga is closed we have

∗ X Ga,χ S (NGa,a) = 0.

Then S∗(X)G,χ = 0.

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Fourier transform

Let V be a ﬁnite dimensional vector space over F and Q be a non-degenerate quadratic form on V . Let ξb denote the Fourier transform of ξ deﬁned using Q. Proposition Let G act on V linearly and preserving Q. Let ξ ∈ S∗(V )G,χ. Then ξb ∈ S∗(V )G,χ.

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Theorem (Homogeneity theorem – Jacquet, Rallis, Schiffmann,...) ∗ Assume F is non-Archimedean. Let ξ ∈ SV (Z(Q)) be s.t. ∗ 1 ξb ∈ SV (Z(Q)). Then ξ is abs-homogeneous of degree 2 dimV .

Fourier transform and homogeneity

We call a distribution ξ ∈ S∗(V ) abs-homogeneous of degree d if for any t ∈ F ×, d ht (ξ) = u(t)|t| ξ,

where ht denotes the homothety action on distributions and u is some unitary character of F ×. Theorem (Archimedean homogeneity theorem – A.-Gourevitch) ∗ Let F be any local ﬁeld. Let L ⊂ SV (Z(Q)) be a non-zero linear subspace s. t. ∀ξ ∈ L we have ξb ∈ L and Qξ ∈ L. Then there exists a non-zero distribution ξ ∈ L which is 1 1 abs-homogeneous of degree 2 dimV or of degree 2 dimV + 1.

Fourier transform and homogeneity

We call a distribution ξ ∈ S∗(V ) abs-homogeneous of degree d if for any t ∈ F ×, d ht (ξ) = u(t)|t| ξ,

where ht denotes the homothety action on distributions and u is some unitary character of F ×. Theorem (Homogeneity theorem – Jacquet, Rallis, Schiffmann,...) ∗ Assume F is non-Archimedean. Let ξ ∈ SV (Z(Q)) be s.t. ∗ 1 ξb ∈ SV (Z(Q)). Then ξ is abs-homogeneous of degree 2 dimV . Fourier transform and homogeneity

We call a distribution ξ ∈ S∗(V ) abs-homogeneous of degree d if for any t ∈ F ×, d ht (ξ) = u(t)|t| ξ,

Theorem (Archimedean homogeneity theorem – A.-Gourevitch) ∗ Let F be any local field. Let L ⊂ SV (Z(Q)) be a non-zero linear subspace s. t. ∀ξ ∈ L we have ξb ∈ L and Qξ ∈ L. Then there exists a non-zero distribution ξ ∈ L which is 1 1 abs-homogeneous of degree 2 dimV or of degree 2 dimV + 1. Singular Support Wave front set (=Characteristic variety) Defined using D-modules Defined using Fourier transform Available only in the Available in both cases Archimedean case In the non-Archimedean case we define the singular support to be the Zariski closure of the wave front set.

Singular Support and Wave Front Set

To a distribution ξ on X one assigns two subsets of T ∗X.

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs In the non-Archimedean case we deﬁne the singular support to be the Zariski closure of the wave front set.

Singular Support and Wave Front Set

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs 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 cases Archimedean case In the non-Archimedean case we define the singular support to be the Zariski closure of the wave front set.

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs 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 closed subvariety, invariant with respect to homotheties. Let ξ ∈ S∗(V ). Suppose that Supp(ξb) ⊂ Z . Then SS(ξ) ⊂ V × Z. Integrability theorem: Let ξ ∈ S∗(X). Then SS(ξ) is (weakly) coisotropic.

Properties and the Integrability Theorem

Let X be a smooth algebraic variety. ∗ Let ξ ∈ S (X). Then Supp(ξ)Zar = pX (SS(ξ)), where ∗ pX : T X → X is the projection.

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Let V be a linear space. Let Z ⊂ V ∗ be a closed subvariety, invariant with respect to homotheties. Let ξ ∈ S∗(V ). Suppose that Supp(ξb) ⊂ Z . Then SS(ξ) ⊂ V × Z. Integrability theorem: Let ξ ∈ S∗(X). Then SS(ξ) is (weakly) coisotropic.

Properties and the Integrability Theorem

Let X be a smooth algebraic variety. ∗ Let ξ ∈ S (X). Then Supp(ξ)Zar = pX (SS(ξ)), where ∗ pX : 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}.

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Integrability theorem: Let ξ ∈ S∗(X). Then SS(ξ) is (weakly) coisotropic.

Properties and the Integrability Theorem

Let X be a smooth algebraic variety. ∗ Let ξ ∈ S (X). Then Supp(ξ)Zar = pX (SS(ξ)), where ∗ pX : 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 closed subvariety, invariant with respect to homotheties. Let ξ ∈ S∗(V ). Suppose that Supp(ξb) ⊂ Z . Then SS(ξ) ⊂ V × Z.

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Properties and the Integrability Theorem

Let X be a smooth algebraic variety. ∗ Let ξ ∈ S (X). Then Supp(ξ)Zar = pX (SS(ξ)), where ∗ pX : 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}.

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Every non-empty coisotropic subvariety of M has dim M dimension at least 2 .

Coisotropic varieties

Deﬁnition Let M be a smooth algebraic variety and ω be a symplectic form on it. Let Z ⊂ M be an algebraic subvariety. We call it M-coisotropic if the following equivalent conditions hold. ⊥ At every smooth point z ∈ Z we have Tz Z ⊃ (Tz Z ) . Here, ⊥ (Tz Z ) denotes the orthogonal complement to Tz Z in Tz M with respect to ω. The ideal sheaf of regular functions that vanish on Z is closed under Poisson bracket. If there is no ambiguity, we will call Z a coisotropic variety.

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Coisotropic varieties

Every non-empty coisotropic subvariety of M has dim M dimension at least 2 .

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Every non-empty weakly coisotropic subvariety of T ∗X has dimension at least dim X.

Weakly coisotropic varieties

Deﬁnition Let X be a smooth algebraic variety. Let Z ⊂ T ∗X be an algebraic subvariety. We call it T ∗X-weakly coisotropic if one of the following equivalent conditions holds.

For a generic smooth point a ∈ pX (Z) and for a generic −1 smooth point y ∈ pX (a) ∩ Z we have X −1 CN ⊂ Ty (p (a) ∩ Z). pX (Z ),a X −1 For any smooth point a ∈ pX (Z) the ﬁber pX (a) ∩ Z is locally invariant with respect to shifts by CNX . pX (Z ),a

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Weakly coisotropic varieties

Deﬁnition Let X be a smooth algebraic variety. Let Z ⊂ T ∗X be an algebraic subvariety. We call it T ∗X-weakly coisotropic if one of the following equivalent conditions holds.

Every non-empty weakly coisotropic subvariety of T ∗X has dimension at least dim X.

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Lemma Let X be a smooth algebraic variety. Let Z ⊂ X be a smooth subvariety. Let R ⊂ T ∗X be a (weakly) coisotropic variety. ∗ Then, under some transversality assumption, R|Z ⊂ T Z is a (weakly) coisotropic variety.

Deﬁnition Let X be a smooth algebraic variety. Let Z ⊂ X be a smooth subvariety and R ⊂ T ∗X be any subvariety. We deﬁne the ∗ restriction R|Z ⊂ T Z of R to Z by

−1 R|Z := q(pX (Z ) ∩ R),

−1 ∗ where q : pX (Z ) → T Z is the projection.

∗ −1 ∗ T X ⊃ pX (Z ) T Z Deﬁnition Let X be a smooth algebraic variety. Let Z ⊂ X be a smooth subvariety and R ⊂ T ∗X be any subvariety. We deﬁne the ∗ restriction R|Z ⊂ T Z of R to Z by

−1 R|Z := q(pX (Z ) ∩ R),

−1 ∗ where q : pX (Z ) → T Z is the projection.

∗ −1 ∗ T X ⊃ pX (Z ) T Z

Lemma Let X be a smooth algebraic variety. Let Z ⊂ X be a smooth subvariety. Let R ⊂ T ∗X be a (weakly) coisotropic variety. ∗ Then, under some transversality assumption, R|Z ⊂ T Z is a (weakly) coisotropic variety. By Harish-Chandra descent we can assume that any ξ ∈ S∗(X)Ge,χ is supported in S. Notation S0 := {(A, v, φ) ∈ S|An−1v = (A∗)n−1φ = 0}.

By the homogeneity theorem, the stratiﬁcation method and Frobenius descent we get that any ξ ∈ S∗(X)Ge,χ is supported in S0.

Harish-Chandra descent and homogeneity

Notation

n i S := {(A, v, φ) ∈ Xn|A = 0 and φ(A v) = 0 ∀0 ≤ i ≤ n}.

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Notation S0 := {(A, v, φ) ∈ S|An−1v = (A∗)n−1φ = 0}.

By the homogeneity theorem, the stratiﬁcation method and Frobenius descent we get that any ξ ∈ S∗(X)Ge,χ is supported in S0.

Harish-Chandra descent and homogeneity

Notation

n i S := {(A, v, φ) ∈ Xn|A = 0 and φ(A v) = 0 ∀0 ≤ i ≤ n}.

By Harish-Chandra descent we can assume that any ξ ∈ S∗(X)Ge,χ is supported in S.

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs By the homogeneity theorem, the stratiﬁcation method and Frobenius descent we get that any ξ ∈ S∗(X)Ge,χ is supported in S0.

Harish-Chandra descent and homogeneity

Notation

n i S := {(A, v, φ) ∈ Xn|A = 0 and φ(A v) = 0 ∀0 ≤ i ≤ n}.

By Harish-Chandra descent we can assume that any ξ ∈ S∗(X)Ge,χ is supported in S. Notation S0 := {(A, v, φ) ∈ S|An−1v = (A∗)n−1φ = 0}.

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Harish-Chandra descent and homogeneity

Notation

n i S := {(A, v, φ) ∈ Xn|A = 0 and φ(A v) = 0 ∀0 ≤ i ≤ n}.

By Harish-Chandra descent we can assume that any ξ ∈ S∗(X)Ge,χ is supported in S. Notation S0 := {(A, v, φ) ∈ S|An−1v = (A∗)n−1φ = 0}.

By the homogeneity theorem, the stratiﬁcation method and Frobenius descent we get that any ξ ∈ S∗(X)Ge,χ is supported in S0.

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs It is enough to show: Theorem (The geometric statement) There are no non-empty X × X-weakly coisotropic subvarieties of T 0.

Reduction to the geometric statement

Notation

0 T = {((A1, v1, φ1), (A2, v2, φ2)) ∈ X × X | ∀i, j ∈ {1, 2} 0 (Ai , vj , φj ) ∈ S and [A1, A2] + v1 ⊗ φ2 − v2 ⊗ φ1 = 0}.

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Reduction to the geometric statement

Notation

0 T = {((A1, v1, φ1), (A2, v2, φ2)) ∈ X × X | ∀i, j ∈ {1, 2} 0 (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 subvarieties of T 0.

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Summary

Flowchart

H.Ch. Fourier transform and Fourier transform and sl(V ) × V × V ∗ / S / S0 / T 0 descent homogeneity theorem integrability theorem

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs It is easy to see that there are no non-empty X × X-weakly coisotropic subvarieties of T 00. Notation Let A ∈ sl(V ) be a nilpotent Jordan block. Denote 0 00 RA := (T − T )|{A}×V ×V ∗ .

It is enough to show: Lemma (Key Lemma) There are no non-empty V × V ∗ × V × V ∗-weakly coisotropic subvarieties of RA.

Reduction to the Key Lemma

Notation

00 0 n−1 T := {((A1, v1, φ1), (A2, v2, φ2)) ∈ T |A1 = 0}.

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Notation Let A ∈ sl(V ) be a nilpotent Jordan block. Denote 0 00 RA := (T − T )|{A}×V ×V ∗ .

It is enough to show: Lemma (Key Lemma) There are no non-empty V × V ∗ × V × V ∗-weakly coisotropic subvarieties of RA.

Reduction to the Key Lemma

Notation

00 0 n−1 T := {((A1, v1, φ1), (A2, v2, φ2)) ∈ T |A1 = 0}. It is easy to see that there are no non-empty X × X-weakly coisotropic subvarieties of T 00.

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs It is enough to show: Lemma (Key Lemma) There are no non-empty V × V ∗ × V × V ∗-weakly coisotropic subvarieties of RA.

Reduction to the Key Lemma

Notation

00 0 n−1 T := {((A1, v1, φ1), (A2, v2, φ2)) ∈ T |A1 = 0}. It is easy to see that there are no non-empty X × X-weakly coisotropic subvarieties of T 00. Notation Let A ∈ sl(V ) be a nilpotent Jordan block. Denote 0 00 RA := (T − T )|{A}×V ×V ∗ .

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Reduction to the Key Lemma

Notation

It is enough to show: Lemma (Key Lemma) There are no non-empty V × V ∗ × V × V ∗-weakly coisotropic subvarieties of RA.

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Sn−1 It is easy to see that RA ⊂ QA × QA and QA × QA = i,j=1 Lij , where i ∗ n−i j ∗ n−j Lij := (KerA ) × (Ker(A ) ) × (KerA ) × (Ker(A ) ). It is easy to see that any weakly coisotropic subvariety of Sn−1 QA × QA is contained in i=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}.

Proof of the Key Lemma

Notation

n−1 0 ∗ [ i ∗ n−i QA := S ∩ ({A} × V × V ) = (KerA ) × (Ker(A ) ) i=1

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Sn−1 and QA × QA = i,j=1 Lij , where i ∗ n−i j ∗ n−j Lij := (KerA ) × (Ker(A ) ) × (KerA ) × (Ker(A ) ). It is easy to see that any weakly coisotropic subvariety of Sn−1 QA × QA is contained in i=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}.

Proof of the Key Lemma

Notation

n−1 0 ∗ [ i ∗ n−i QA := S ∩ ({A} × V × V ) = (KerA ) × (Ker(A ) ) i=1

It is easy to see that RA ⊂ QA × QA

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs It is easy to see that any weakly coisotropic subvariety of Sn−1 QA × QA is contained in i=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}.

Proof of the Key Lemma

Notation

n−1 0 ∗ [ i ∗ n−i QA := S ∩ ({A} × V × V ) = (KerA ) × (Ker(A ) ) i=1 Sn−1 It is easy to see that RA ⊂ QA × QA and QA × QA = i,j=1 Lij , where i ∗ n−i j ∗ n−j Lij := (KerA ) × (Ker(A ) ) × (KerA ) × (Ker(A ) ).

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs 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}.

Proof of the Key Lemma

Notation

n−1 0 ∗ [ i ∗ n−i QA := S ∩ ({A} × V × V ) = (KerA ) × (Ker(A ) ) i=1 Sn−1 It is easy to see that RA ⊂ QA × QA and QA × QA = i,j=1 Lij , where i ∗ n−i j ∗ n−j Lij := (KerA ) × (Ker(A ) ) × (KerA ) × (Ker(A ) ). It is easy to see that any weakly coisotropic subvariety of Sn−1 QA × QA is contained in i=1 Lii .

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs 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}.

Proof of the Key Lemma

Notation

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs It is enough to show that f (RA ∩ Lii ) = {0}.

Proof of the Key Lemma

Notation

f (v1, φ1, v2, φ2) := (v1)i (φ2)i+1 − (v2)i (φ1)i+1.

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Proof of the Key Lemma

Notation

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}.

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Clearly, M is of the form

0 ∗ M = i×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 and hence f (v1, φ1, v2, φ2) = 0.

Proof of the Key Lemma

Let (v1, φ1, v2, φ2) ∈ RA ∩ Lii . Let M := v1 ⊗ φ2 − v2 ⊗ φ1.

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs We know that there exists a nilpotent B satisfying [A, B] = M. Hence this B is upper nilpotent, which implies Mi,i+1 = 0 and hence f (v1, φ1, v2, φ2) = 0.

Proof of the Key Lemma

Let (v1, φ1, v2, φ2) ∈ RA ∩ Lii . Let M := v1 ⊗ φ2 − v2 ⊗ φ1. Clearly, M is of the form

0 ∗ M = i×i . 0(n−i)×i 0(n−i)×(n−i)

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Hence this B is upper nilpotent, which implies Mi,i+1 = 0 and hence f (v1, φ1, v2, φ2) = 0.

Proof of the Key Lemma

Let (v1, φ1, v2, φ2) ∈ RA ∩ Lii . Let M := v1 ⊗ φ2 − v2 ⊗ φ1. Clearly, M is of the form

0 ∗ M = i×i . 0(n−i)×i 0(n−i)×(n−i)

We know that there exists a nilpotent B satisfying [A, B] = M.

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs and hence f (v1, φ1, v2, φ2) = 0.

Proof of the Key Lemma

Let (v1, φ1, v2, φ2) ∈ RA ∩ Lii . Let M := v1 ⊗ φ2 − v2 ⊗ φ1. Clearly, M is of the form

0 ∗ M = i×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

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Proof of the Key Lemma

Let (v1, φ1, v2, φ2) ∈ RA ∩ Lii . Let M := v1 ⊗ φ2 − v2 ⊗ φ1. Clearly, M is of the form

0 ∗ M = i×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 and hence f (v1, φ1, v2, φ2) = 0.

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Summary

Flowchart

sl(V ) × V × V ∗

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Summary

Flowchart

H.Ch. sl(V ) × V × V ∗ / S descent

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Summary

Flowchart

H.Ch. Fourier transform and sl(V ) × V × V ∗ / S / S0 descent homogeneity theorem

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Summary

Flowchart

H.Ch. Fourier transform and Fourier transform and sl(V ) × V × V ∗ / S / S0 / T 0 descent homogeneity theorem integrability theorem

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Summary

Flowchart

H.Ch. Fourier transform and Fourier transform and sl(V ) × V × V ∗ / S / S0 / T 0 descent homogeneity theorem integrability theorem

T 0 − T 00

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Summary

Flowchart

H.Ch. Fourier transform and Fourier transform and sl(V ) × V × V ∗ / S / S0 / T 0 descent homogeneity theorem integrability theorem

restriction 0 00 RA o T − T

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Summary

Flowchart

H.Ch. Fourier transform and Fourier transform and sl(V ) × V × V ∗ / S / S0 / T 0 descent homogeneity theorem integrability theorem

R ⊂S L A ij restriction 0 00 Lii ∩ RA o RA o T − T

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Summary

Flowchart

H.Ch. Fourier transform and Fourier transform and sl(V ) × V × V ∗ / S / S0 / T 0 descent homogeneity theorem integrability theorem

f (R ∩L )=0 R ⊂S L A ii A ij restriction 0 00 ∅ o Lii ∩ RA o RA o T − T

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs We want to prove:

S∗(X)G,χ = 0.

Applications: Representation theory, Harmonic analysis, Gelfand pairs, trace formula, relative trace formula, ...

Necessary condition:

Close orbits do not carry equivariant distributions m

χ|Ga 6= 1 for any semi simple a ∈ X (i.e. a ∈ X with closed orbit Ga)

Aim

Task Let a reductive group G act on an afﬁne variety X and let χ be a character of G.

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Applications: Representation theory, Harmonic analysis, Gelfand pairs, trace formula, relative trace formula, ...

Necessary condition:

Close orbits do not carry equivariant distributions m

χ|Ga 6= 1 for any semi simple a ∈ X (i.e. a ∈ X with closed orbit Ga)

Aim

Task Let a reductive group G act on an afﬁne variety X and let χ be a character of G. We want to prove:

S∗(X)G,χ = 0.

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Representation theory, Harmonic analysis, Gelfand pairs, trace formula, relative trace formula, ...

Necessary condition:

Close orbits do not carry equivariant distributions m

χ|Ga 6= 1 for any semi simple a ∈ X (i.e. a ∈ X with closed orbit Ga)

Aim

Task Let a reductive group G act on an afﬁne variety X and let χ be a character of G. We want to prove:

S∗(X)G,χ = 0.

Applications:

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Gelfand pairs, trace formula, relative trace formula, ...

Necessary condition:

Close orbits do not carry equivariant distributions m

χ|Ga 6= 1 for any semi simple a ∈ X (i.e. a ∈ X with closed orbit Ga)

Aim

Task Let a reductive group G act on an afﬁne variety X and let χ be a character of G. We want to prove:

S∗(X)G,χ = 0.

Applications: Representation theory, Harmonic analysis,

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs trace formula, relative trace formula, ...

Necessary condition:

Close orbits do not carry equivariant distributions m

χ|Ga 6= 1 for any semi simple a ∈ X (i.e. a ∈ X with closed orbit Ga)

Aim

Task Let a reductive group G act on an afﬁne variety X and let χ be a character of G. We want to prove:

S∗(X)G,χ = 0.

Applications: Representation theory, Harmonic analysis, Gelfand pairs,

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Necessary condition:

Close orbits do not carry equivariant distributions m

χ|Ga 6= 1 for any semi simple a ∈ X (i.e. a ∈ X with closed orbit Ga)

Aim

Task Let a reductive group G act on an afﬁne variety X and let χ be a character of G. We want to prove:

S∗(X)G,χ = 0.

Applications: Representation theory, Harmonic analysis, Gelfand pairs, trace formula, relative trace formula, ...

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs m

χ|Ga 6= 1 for any semi simple a ∈ X (i.e. a ∈ X with closed orbit Ga)

Aim

Task Let a reductive group G act on an afﬁne variety X and let χ be a character of G. We want to prove:

S∗(X)G,χ = 0.

Applications: Representation theory, Harmonic analysis, Gelfand pairs, trace formula, relative trace formula, ...

Necessary condition:

Close orbits do not carry equivariant distributions

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Aim

Task Let a reductive group G act on an afﬁne variety X and let χ be a character of G. We want to prove:

S∗(X)G,χ = 0.

Applications: Representation theory, Harmonic analysis, Gelfand pairs, trace formula, relative trace formula, ...

Necessary condition:

Close orbits do not carry equivariant distributions m

χ|Ga 6= 1 for any semi simple a ∈ X (i.e. a ∈ X with closed orbit Ga)

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Theorem (Luna) Let a reductive group G act on a smooth afﬁne algebraic variety X. Let a ∈ X be a semi-simple point. Then there exists an invariant (etale) neighborhood U of Ga with an equivariant projection p : U → Ga s.t. the ﬁber p−1(a) is G-isomorphic to X an (etale) neighborhood of 0 in the normal space NGa,a.

Luna’s slice theorem

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Luna’s slice theorem

Theorem (Luna) Let a reductive group G act on a smooth afﬁne algebraic variety X. Let a ∈ X be a semi-simple point. Then there exists an invariant (etale) neighborhood U of Ga with an equivariant projection p : U → Ga s.t. the ﬁber p−1(a) is G-isomorphic to X an (etale) neighborhood of 0 in the normal space NGa,a.

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Generalized Harish-Chandra descent

Theorem (A.-Gourevitch) Let a reductive group G act on a smooth afﬁne algebraic variety X. Let χ be a character of G. Suppose that for any a ∈ X s.t. the orbit Ga is closed, we have

∗ X Ga,χ S (NGa,a) = 0.

Then S∗(X)G,χ = 0.

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs We reduce to the following Task Let a reductive group G act (linearly) on an linear space V and let χ be a character of G. We should prove that

S∗(V )G,χ = 0

We may assume V G = 0 Let p : V → V //G := spec(O(V )G) Let N (V ) := p−1(p(0)) = {x ∈ V |Gx 3 0} Let R(V ) := V − N (V )

by induction we may assume: S∗(R(V ))G,χ = 0.

Conclusions

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs We should prove that

S∗(V )G,χ = 0

We may assume V G = 0 Let p : V → V //G := spec(O(V )G) Let N (V ) := p−1(p(0)) = {x ∈ V |Gx 3 0} Let R(V ) := V − N (V )

by induction we may assume: S∗(R(V ))G,χ = 0.

Conclusions

We reduce to the following Task Let a reductive group G act (linearly) on an linear space V and let χ be a character of G.

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs We may assume V G = 0 Let p : V → V //G := spec(O(V )G) Let N (V ) := p−1(p(0)) = {x ∈ V |Gx 3 0} Let R(V ) := V − N (V )

by induction we may assume: S∗(R(V ))G,χ = 0.

Conclusions

We reduce to the following Task Let a reductive group G act (linearly) on an linear space V and let χ be a character of G. We should prove that

S∗(V )G,χ = 0

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Let p : V → V //G := spec(O(V )G) Let N (V ) := p−1(p(0)) = {x ∈ V |Gx 3 0} Let R(V ) := V − N (V )

by induction we may assume: S∗(R(V ))G,χ = 0.

Conclusions

We reduce to the following Task Let a reductive group G act (linearly) on an linear space V and let χ be a character of G. We should prove that

S∗(V )G,χ = 0

We may assume V G = 0

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Let N (V ) := p−1(p(0)) = {x ∈ V |Gx 3 0} Let R(V ) := V − N (V )

by induction we may assume: S∗(R(V ))G,χ = 0.

Conclusions

We reduce to the following Task Let a reductive group G act (linearly) on an linear space V and let χ be a character of G. We should prove that

S∗(V )G,χ = 0

We may assume V G = 0 Let p : V → V //G := spec(O(V )G)

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs = {x ∈ V |Gx 3 0} Let R(V ) := V − N (V )

by induction we may assume: S∗(R(V ))G,χ = 0.

Conclusions

We reduce to the following Task Let a reductive group G act (linearly) on an linear space V and let χ be a character of G. We should prove that

S∗(V )G,χ = 0

We may assume V G = 0 Let p : V → V //G := spec(O(V )G) Let N (V ) := p−1(p(0))

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Let R(V ) := V − N (V )

by induction we may assume: S∗(R(V ))G,χ = 0.

Conclusions

We reduce to the following Task Let a reductive group G act (linearly) on an linear space V and let χ be a character of G. We should prove that

S∗(V )G,χ = 0

We may assume V G = 0 Let p : V → V //G := spec(O(V )G) Let N (V ) := p−1(p(0)) = {x ∈ V |Gx 3 0}

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs by induction we may assume: S∗(R(V ))G,χ = 0.

Conclusions

We reduce to the following Task Let a reductive group G act (linearly) on an linear space V and let χ be a character of G. We should prove that

S∗(V )G,χ = 0

We may assume V G = 0 Let p : V → V //G := spec(O(V )G) Let N (V ) := p−1(p(0)) = {x ∈ V |Gx 3 0} Let R(V ) := V − N (V )

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Conclusions

We reduce to the following Task Let a reductive group G act (linearly) on an linear space V and let χ be a character of G. We should prove that

S∗(V )G,χ = 0

We may assume V G = 0 Let p : V → V //G := spec(O(V )G) Let N (V ) := p−1(p(0)) = {x ∈ V |Gx 3 0} Let R(V ) := V − N (V )

by induction we may assume: S∗(R(V ))G,χ = 0.

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Using Fourier transform, we reduce the problem to showing Task

∗ ∗ G,χ (SN (V )(V ) ∩ F(SN (V )(V )) = 0

Using homogeneity theorem, we reduce the problem to showing Task

∗ ∗ G×F ×,χ×u (SN (V )(V ) ∩ F(SN (V )(V )) = 0

Conclusions

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Task

∗ ∗ G,χ (SN (V )(V ) ∩ F(SN (V )(V )) = 0

Using homogeneity theorem, we reduce the problem to showing Task

∗ ∗ G×F ×,χ×u (SN (V )(V ) ∩ F(SN (V )(V )) = 0

Conclusions

Using Fourier transform, we reduce the problem to showing

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Using homogeneity theorem, we reduce the problem to showing Task

∗ ∗ G×F ×,χ×u (SN (V )(V ) ∩ F(SN (V )(V )) = 0

Conclusions

Using Fourier transform, we reduce the problem to showing Task

∗ ∗ G,χ (SN (V )(V ) ∩ F(SN (V )(V )) = 0

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Conclusions

Using Fourier transform, we reduce the problem to showing Task

∗ ∗ G,χ (SN (V )(V ) ∩ F(SN (V )(V )) = 0

Using homogeneity theorem, we reduce the problem to showing Task

∗ ∗ G×F ×,χ×u (SN (V )(V ) ∩ F(SN (V )(V )) = 0

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs A symmetric pair is a triple (G, H, θ) where H ⊂ G are reductive groups, and θ is an involution of G such that H = Gθ.

We call (G, H, θ) connected if G/H is Zariski connected.

Deﬁne an anti-involution σ : G → G by σ(g) := θ(g−1).

Symmetric pairs

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs We call (G, H, θ) connected if G/H is Zariski connected.

Deﬁne an anti-involution σ : G → G by σ(g) := θ(g−1).

Symmetric pairs

A symmetric pair is a triple (G, H, θ) where H ⊂ G are reductive groups, and θ is an involution of G such that H = Gθ.

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Deﬁne an anti-involution σ : G → G by σ(g) := θ(g−1).

Symmetric pairs

A symmetric pair is a triple (G, H, θ) where H ⊂ G are reductive groups, and θ is an involution of G such that H = Gθ.

We call (G, H, θ) connected if G/H is Zariski connected.

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Symmetric pairs

A symmetric pair is a triple (G, H, θ) where H ⊂ G are reductive groups, and θ is an involution of G such that H = Gθ.

We call (G, H, θ) connected if G/H is Zariski connected.

Deﬁne an anti-involution σ : G → G by σ(g) := θ(g−1).

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs For symmetric pairs of rank one this question was studied extensively by van-Dijk, Bosman, Rader and Rallis. Task S∗(G)H×H ⊂ S∗(G)σ

Necessary condition: Deﬁnition A symmetric pair (G, H, θ) is called good if σ preserves all closed H × H double closets.

Proposition Any connected symmetric pair over C is good.

Conjecture Any good symmetric pair is a Gelfand pair.

Question Which symmetric pairs are Gelfand pairs? Task S∗(G)H×H ⊂ S∗(G)σ

Necessary condition: Deﬁnition A symmetric pair (G, H, θ) is called good if σ preserves all closed H × H double closets.

Proposition Any connected symmetric pair over C is good.

Conjecture Any good symmetric pair is a Gelfand pair.

Question Which symmetric pairs are Gelfand pairs?

For symmetric pairs of rank one this question was studied extensively by van-Dijk, Bosman, Rader and Rallis. Necessary condition: Deﬁnition A symmetric pair (G, H, θ) is called good if σ preserves all closed H × H double closets.

Proposition Any connected symmetric pair over C is good.

Conjecture Any good symmetric pair is a Gelfand pair.

Question Which symmetric pairs are Gelfand pairs?

For symmetric pairs of rank one this question was studied extensively by van-Dijk, Bosman, Rader and Rallis. Task S∗(G)H×H ⊂ S∗(G)σ Proposition Any connected symmetric pair over C is good.

Conjecture Any good symmetric pair is a Gelfand pair.

Question Which symmetric pairs are Gelfand pairs?

For symmetric pairs of rank one this question was studied extensively by van-Dijk, Bosman, Rader and Rallis. Task S∗(G)H×H ⊂ S∗(G)σ

Necessary condition: Deﬁnition A symmetric pair (G, H, θ) is called good if σ preserves all closed H × H double closets. Conjecture Any good symmetric pair is a Gelfand pair.

Question Which symmetric pairs are Gelfand pairs?

For symmetric pairs of rank one this question was studied extensively by van-Dijk, Bosman, Rader and Rallis. Task S∗(G)H×H ⊂ S∗(G)σ

Necessary condition: Deﬁnition A symmetric pair (G, H, θ) is called good if σ preserves all closed H × H double closets.

Proposition Any connected symmetric pair over C is good. Question Which symmetric pairs are Gelfand pairs?

For symmetric pairs of rank one this question was studied extensively by van-Dijk, Bosman, Rader and Rallis. Task S∗(G)H×H ⊂ S∗(G)σ

Necessary condition: Deﬁnition A symmetric pair (G, H, θ) is called good if σ preserves all closed H × H double closets.

Proposition Any connected symmetric pair over C is good.

Conjecture Any good symmetric pair is a Gelfand pair. Reformulating our task Task

Let H^× H = H × H o {1, σ} and χ : H^× H → C deﬁned by χ(H^× H − H × H) = −1 we have to show that

S∗(G)H]×H,χ = 0

Using Harsh-Chandra Descent it is enough to show that 1 The pair (G, H) is good

2 S∗(gσ)He,χ = 0 provided that S∗(R(gσ))He,χ = 0.

3 Compute all the "descendants" of the pair and prove (2) for them. We call the property (2) regularity. We conjecture that all symmetric pairs are regular. This will imply that any good symmetric pair is a Gelfand pair.

How to complete the task? Task

Let H^× H = H × H o {1, σ} and χ : H^× H → C deﬁned by χ(H^× H − H × H) = −1 we have to show that

S∗(G)H]×H,χ = 0

Using Harsh-Chandra Descent it is enough to show that 1 The pair (G, H) is good

2 S∗(gσ)He,χ = 0 provided that S∗(R(gσ))He,χ = 0.

How to complete the task?

Reformulating our task Let H^× H = H × H o {1, σ} and χ : H^× H → C deﬁned by χ(H^× H − H × H) = −1 we have to show that

S∗(G)H]×H,χ = 0

Using Harsh-Chandra Descent it is enough to show that 1 The pair (G, H) is good

2 S∗(gσ)He,χ = 0 provided that S∗(R(gσ))He,χ = 0.

How to complete the task?

Reformulating our task Task and χ : H^× H → C deﬁned by χ(H^× H − H × H) = −1 we have to show that

S∗(G)H]×H,χ = 0

Using Harsh-Chandra Descent it is enough to show that 1 The pair (G, H) is good

2 S∗(gσ)He,χ = 0 provided that S∗(R(gσ))He,χ = 0.

How to complete the task?

Reformulating our task Task

Let H^× H = H × H o {1, σ} deﬁned by χ(H^× H − H × H) = −1 we have to show that

S∗(G)H]×H,χ = 0

Using Harsh-Chandra Descent it is enough to show that 1 The pair (G, H) is good

2 S∗(gσ)He,χ = 0 provided that S∗(R(gσ))He,χ = 0.

How to complete the task?

Reformulating our task Task

Let H^× H = H × H o {1, σ} and χ : H^× H → C we have to show that

S∗(G)H]×H,χ = 0

Using Harsh-Chandra Descent it is enough to show that 1 The pair (G, H) is good

2 S∗(gσ)He,χ = 0 provided that S∗(R(gσ))He,χ = 0.

How to complete the task?

Reformulating our task Task

Let H^× H = H × H o {1, σ} and χ : H^× H → C deﬁned by χ(H^× H − H × H) = −1 S∗(G)H]×H,χ = 0

Using Harsh-Chandra Descent it is enough to show that 1 The pair (G, H) is good

2 S∗(gσ)He,χ = 0 provided that S∗(R(gσ))He,χ = 0.

How to complete the task?

Reformulating our task Task

Let H^× H = H × H o {1, σ} and χ : H^× H → C deﬁned by χ(H^× H − H × H) = −1 we have to show that Using Harsh-Chandra Descent it is enough to show that 1 The pair (G, H) is good

2 S∗(gσ)He,χ = 0 provided that S∗(R(gσ))He,χ = 0.

How to complete the task?

Reformulating our task Task

Let H^× H = H × H o {1, σ} and χ : H^× H → C deﬁned by χ(H^× H − H × H) = −1 we have to show that

S∗(G)H]×H,χ = 0 1 The pair (G, H) is good

2 S∗(gσ)He,χ = 0 provided that S∗(R(gσ))He,χ = 0.

How to complete the task?

Reformulating our task Task

Let H^× H = H × H o {1, σ} and χ : H^× H → C deﬁned by χ(H^× H − H × H) = −1 we have to show that

S∗(G)H]×H,χ = 0

Using Harsh-Chandra Descent it is enough to show that 2 S∗(gσ)He,χ = 0 provided that S∗(R(gσ))He,χ = 0.

How to complete the task?

Reformulating our task Task

Let H^× H = H × H o {1, σ} and χ : H^× H → C deﬁned by χ(H^× H − H × H) = −1 we have to show that

S∗(G)H]×H,χ = 0

Using Harsh-Chandra Descent it is enough to show that 1 The pair (G, H) is good 3 Compute all the "descendants" of the pair and prove (2) for them. We call the property (2) regularity. We conjecture that all symmetric pairs are regular. This will imply that any good symmetric pair is a Gelfand pair.

How to complete the task?

Reformulating our task Task

Let H^× H = H × H o {1, σ} and χ : H^× H → C deﬁned by χ(H^× H − H × H) = −1 we have to show that

S∗(G)H]×H,χ = 0

Using Harsh-Chandra Descent it is enough to show that 1 The pair (G, H) is good

2 S∗(gσ)He,χ = 0 provided that S∗(R(gσ))He,χ = 0. We call the property (2) regularity. We conjecture that all symmetric pairs are regular. This will imply that any good symmetric pair is a Gelfand pair.

How to complete the task?

Reformulating our task Task

Let H^× H = H × H o {1, σ} and χ : H^× H → C deﬁned by χ(H^× H − H × H) = −1 we have to show that

S∗(G)H]×H,χ = 0

Using Harsh-Chandra Descent it is enough to show that 1 The pair (G, H) is good

2 S∗(gσ)He,χ = 0 provided that S∗(R(gσ))He,χ = 0.

3 Compute all the "descendants" of the pair and prove (2) for them. How to complete the task?

Reformulating our task Task

Let H^× H = H × H o {1, σ} and χ : H^× H → C deﬁned by χ(H^× H − H × H) = −1 we have to show that

S∗(G)H]×H,χ = 0

Using Harsh-Chandra Descent it is enough to show that 1 The pair (G, H) is good

2 S∗(gσ)He,χ = 0 provided that S∗(R(gσ))He,χ = 0.

σ Let H0 = He × F × and χ0 = χ × | · |dim(g )/2(+1)u Using Homogeneity theorem it is enough to prove that: ∗ σ H0,χ0 ∗ σ H0,χ0 (Sgσ (N (g )) ) ∩ F(Sgσ (N (g )) ) = 0 σ We call an element a ∈ N (g ) distinguished if ha is nilpotent. Using Integrability theorem it is enough to prove that: ∗ H0,χ0 Sgσ (O) = 0 for any distinguished orbit O Using Frobenius descent it is enough to prove that for any distinguished a:

0 0 χ |Ha ∆ 6= 1 – in the non-Archimedean case

gσ 0 0 Ha,χ ∆ (NHa,a) = 0 – in the Archimedean case

How to prove regularity?

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs σ Let H0 = He × F × and χ0 = χ × | · |dim(g )/2(+1)u Using Homogeneity theorem it is enough to prove that: ∗ σ H0,χ0 ∗ σ H0,χ0 (Sgσ (N (g )) ) ∩ F(Sgσ (N (g )) ) = 0 σ We call an element a ∈ N (g ) distinguished if ha is nilpotent. Using Integrability theorem it is enough to prove that: ∗ H0,χ0 Sgσ (O) = 0 for any distinguished orbit O Using Frobenius descent it is enough to prove that for any distinguished a:

0 0 χ |Ha ∆ 6= 1 – in the non-Archimedean case

gσ 0 0 Ha,χ ∆ (NHa,a) = 0 – in the Archimedean case

How to prove regularity?

∗ σ ∗ σ He,χ it is enough to prove that (Sgσ (N (g )) ∩ F(Sgσ (N (g )) = 0

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Using Homogeneity theorem it is enough to prove that: ∗ σ H0,χ0 ∗ σ H0,χ0 (Sgσ (N (g )) ) ∩ F(Sgσ (N (g )) ) = 0 σ We call an element a ∈ N (g ) distinguished if ha is nilpotent. Using Integrability theorem it is enough to prove that: ∗ H0,χ0 Sgσ (O) = 0 for any distinguished orbit O Using Frobenius descent it is enough to prove that for any distinguished a:

0 0 χ |Ha ∆ 6= 1 – in the non-Archimedean case

gσ 0 0 Ha,χ ∆ (NHa,a) = 0 – in the Archimedean case

How to prove regularity?

∗ σ ∗ σ He,χ it is enough to prove that (Sgσ (N (g )) ∩ F(Sgσ (N (g )) = 0

σ Let H0 = He × F × and χ0 = χ × | · |dim(g )/2(+1)u

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs σ We call an element a ∈ N (g ) distinguished if ha is nilpotent. Using Integrability theorem it is enough to prove that: ∗ H0,χ0 Sgσ (O) = 0 for any distinguished orbit O Using Frobenius descent it is enough to prove that for any distinguished a:

0 0 χ |Ha ∆ 6= 1 – in the non-Archimedean case

gσ 0 0 Ha,χ ∆ (NHa,a) = 0 – in the Archimedean case

How to prove regularity?

∗ σ ∗ σ He,χ it is enough to prove that (Sgσ (N (g )) ∩ F(Sgσ (N (g )) = 0

σ Let H0 = He × F × and χ0 = χ × | · |dim(g )/2(+1)u Using Homogeneity theorem it is enough to prove that: ∗ σ H0,χ0 ∗ σ H0,χ0 (Sgσ (N (g )) ) ∩ F(Sgσ (N (g )) ) = 0

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Using Integrability theorem it is enough to prove that: ∗ H0,χ0 Sgσ (O) = 0 for any distinguished orbit O Using Frobenius descent it is enough to prove that for any distinguished a:

0 0 χ |Ha ∆ 6= 1 – in the non-Archimedean case

gσ 0 0 Ha,χ ∆ (NHa,a) = 0 – in the Archimedean case

How to prove regularity?

∗ σ ∗ σ He,χ it is enough to prove that (Sgσ (N (g )) ∩ F(Sgσ (N (g )) = 0

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Using Frobenius descent it is enough to prove that for any distinguished a:

0 0 χ |Ha ∆ 6= 1 – in the non-Archimedean case

gσ 0 0 Ha,χ ∆ (NHa,a) = 0 – in the Archimedean case

How to prove regularity?

∗ σ ∗ σ He,χ it is enough to prove that (Sgσ (N (g )) ∩ F(Sgσ (N (g )) = 0

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs 0 0 χ |Ha ∆ 6= 1 – in the non-Archimedean case

gσ 0 0 Ha,χ ∆ (NHa,a) = 0 – in the Archimedean case

How to prove regularity?

∗ σ ∗ σ He,χ it is enough to prove that (Sgσ (N (g )) ∩ F(Sgσ (N (g )) = 0

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs gσ 0 0 Ha,χ ∆ (NHa,a) = 0 – in the Archimedean case

How to prove regularity?

∗ σ ∗ σ He,χ it is enough to prove that (Sgσ (N (g )) ∩ F(Sgσ (N (g )) = 0

0 0 χ |Ha ∆ 6= 1 – in the non-Archimedean case

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs How to prove regularity?

∗ σ ∗ σ He,χ it is enough to prove that (Sgσ (N (g )) ∩ F(Sgσ (N (g )) = 0

0 0 χ |Ha ∆ 6= 1 – in the non-Archimedean case

gσ 0 0 Ha,χ ∆ (NHa,a) = 0 – in the Archimedean case

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs How to prove regularity?

∗ σ ∗ σ He,χ it is enough to prove that (Sgσ (N (g )) ∩ F(Sgσ (N (g )) = 0

0 0 χ |Ha ∆ 6= 1 – in the non-Archimedean case

gσ 0 0 Ha,χ ∆ (NHa,a) = 0 – in the Archimedean case

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Regular symmetric pairs

Pair p-adic case by real case by (G × G, ∆G) A.-Gourevitch (GLn(E), GLn(F)) Flicker (GLn+k , GLn × GLk ) Jacquet-Rallis A.- (On+k , On × Ok ) A.-Gourevitch Gourevitch (GLn, On) (GL2n, Sp2n) Heumos - Rallis A.-Sayag (sp2m, slm ⊕ ga) (e6, sp8) (e6, sl6 ⊕ sl2) Sayag (e7, sl8) A. (based on (e8, so16) work of Sekiguchi) (f4, sp6 ⊕ sl2) (g2, sl2 ⊕ sl2)

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs (SO(3, R), SO(2, R) is a Gelfand pair - spherical harmonics.

Gelfand-Zeitlin basis: (Sn, Sn−1) is a strong Gelfand pair - basis for irreducible representations of Sn The same for O(n, R) and U(n, R). Classiﬁcation of representations: (GL(n, R), O(n, R)) is a Gelfand pair - the irreducible representations of GL(n, R) which have an O(n, R)-invariant vector are the same as characters of the algebra C(O(n, R)\GL(n, R)/O(n, R). The same for the pair (GL(n, C), U(n)).

Some classical applications

Harmonic analysis.

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Gelfand-Zeitlin basis: (Sn, Sn−1) is a strong Gelfand pair - basis for irreducible representations of Sn The same for O(n, R) and U(n, R). Classiﬁcation of representations: (GL(n, R), O(n, R)) is a Gelfand pair - the irreducible representations of GL(n, R) which have an O(n, R)-invariant vector are the same as characters of the algebra C(O(n, R)\GL(n, R)/O(n, R). The same for the pair (GL(n, C), U(n)).

Some classical applications

Harmonic analysis. (SO(3, R), SO(2, R) is a Gelfand pair - spherical harmonics.

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs (Sn, Sn−1) is a strong Gelfand pair - basis for irreducible representations of Sn The same for O(n, R) and U(n, R). Classiﬁcation of representations: (GL(n, R), O(n, R)) is a Gelfand pair - the irreducible representations of GL(n, R) which have an O(n, R)-invariant vector are the same as characters of the algebra C(O(n, R)\GL(n, R)/O(n, R). The same for the pair (GL(n, C), U(n)).

Some classical applications

Harmonic analysis. (SO(3, R), SO(2, R) is a Gelfand pair - spherical harmonics.

Gelfand-Zeitlin basis:

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs The same for O(n, R) and U(n, R). Classiﬁcation of representations: (GL(n, R), O(n, R)) is a Gelfand pair - the irreducible representations of GL(n, R) which have an O(n, R)-invariant vector are the same as characters of the algebra C(O(n, R)\GL(n, R)/O(n, R). The same for the pair (GL(n, C), U(n)).

Some classical applications

Harmonic analysis. (SO(3, R), SO(2, R) is a Gelfand pair - spherical harmonics.

Gelfand-Zeitlin basis: (Sn, Sn−1) is a strong Gelfand pair - basis for irreducible representations of Sn

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Classiﬁcation of representations: (GL(n, R), O(n, R)) is a Gelfand pair - the irreducible representations of GL(n, R) which have an O(n, R)-invariant vector are the same as characters of the algebra C(O(n, R)\GL(n, R)/O(n, R). The same for the pair (GL(n, C), U(n)).

Some classical applications

Harmonic analysis. (SO(3, R), SO(2, R) is a Gelfand pair - spherical harmonics.

Gelfand-Zeitlin basis: (Sn, Sn−1) is a strong Gelfand pair - basis for irreducible representations of Sn The same for O(n, R) and U(n, R).

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs (GL(n, R), O(n, R)) is a Gelfand pair - the irreducible representations of GL(n, R) which have an O(n, R)-invariant vector are the same as characters of the algebra C(O(n, R)\GL(n, R)/O(n, R). The same for the pair (GL(n, C), U(n)).

Some classical applications

Harmonic analysis. (SO(3, R), SO(2, R) is a Gelfand pair - spherical harmonics.

Gelfand-Zeitlin basis: (Sn, Sn−1) is a strong Gelfand pair - basis for irreducible representations of Sn The same for O(n, R) and U(n, R). Classiﬁcation of representations:

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs The same for the pair (GL(n, C), U(n)).

Some classical applications

Harmonic analysis. (SO(3, R), SO(2, R) is a Gelfand pair - spherical harmonics.

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Some classical applications

Harmonic analysis. (SO(3, R), SO(2, R) is a Gelfand pair - spherical harmonics.

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Automorphic forms

Automorphic multiplicity one ⇔ (GLn(A), GLn(Q)) is a Gelfand pair ⇐ (GLn, Un, ψ) is a Gelfand pair. splitting of automorphic periods: Automorphic periods – integrals of automorphic forms over a subgroup H. (G, H) is a Gelfand pair ⇒ H-period splits into local factors.

More modern applications

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Automorphic multiplicity one ⇔ (GLn(A), GLn(Q)) is a Gelfand pair ⇐ (GLn, Un, ψ) is a Gelfand pair. splitting of automorphic periods: Automorphic periods – integrals of automorphic forms over a subgroup H. (G, H) is a Gelfand pair ⇒ H-period splits into local factors.

More modern applications

Automorphic forms

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs ⇔ (GLn(A), GLn(Q)) is a Gelfand pair ⇐ (GLn, Un, ψ) is a Gelfand pair. splitting of automorphic periods: Automorphic periods – integrals of automorphic forms over a subgroup H. (G, H) is a Gelfand pair ⇒ H-period splits into local factors.

More modern applications

Automorphic forms

Automorphic multiplicity one

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs ⇐ (GLn, Un, ψ) is a Gelfand pair. splitting of automorphic periods: Automorphic periods – integrals of automorphic forms over a subgroup H. (G, H) is a Gelfand pair ⇒ H-period splits into local factors.

More modern applications

Automorphic forms

Automorphic multiplicity one ⇔ (GLn(A), GLn(Q)) is a Gelfand pair

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs splitting of automorphic periods: Automorphic periods – integrals of automorphic forms over a subgroup H. (G, H) is a Gelfand pair ⇒ H-period splits into local factors.

More modern applications

Automorphic forms

Automorphic multiplicity one ⇔ (GLn(A), GLn(Q)) is a Gelfand pair ⇐ (GLn, Un, ψ) is a Gelfand pair.

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Automorphic periods – integrals of automorphic forms over a subgroup H. (G, H) is a Gelfand pair ⇒ H-period splits into local factors.

More modern applications

Automorphic forms

Automorphic multiplicity one ⇔ (GLn(A), GLn(Q)) is a Gelfand pair ⇐ (GLn, Un, ψ) is a Gelfand pair. splitting of automorphic periods:

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs (G, H) is a Gelfand pair ⇒ H-period splits into local factors.

More modern applications

Automorphic forms

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs More modern applications

Automorphic forms

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Study of the principal series

Kirillov conjecture: Any irreducible unitary representation of GLn remains irreducible when restricted to Pn ⇑ any Ad(Pn) invariant distribution on GLn is Ad(GLn) invariant ⇑ any Ad(GLn−1) invariant distribution on GLn is transposition invariant

Trace formula and relative trace formula: Smooth matching

Other applications of invariant distributions

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Kirillov conjecture: Any irreducible unitary representation of GLn remains irreducible when restricted to Pn ⇑ any Ad(Pn) invariant distribution on GLn is Ad(GLn) invariant ⇑ any Ad(GLn−1) invariant distribution on GLn is transposition invariant

Trace formula and relative trace formula: Smooth matching

Other applications of invariant distributions

Study of the principal series

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Any irreducible unitary representation of GLn remains irreducible when restricted to Pn ⇑ any Ad(Pn) invariant distribution on GLn is Ad(GLn) invariant ⇑ any Ad(GLn−1) invariant distribution on GLn is transposition invariant

Trace formula and relative trace formula: Smooth matching

Other applications of invariant distributions

Study of the principal series

Kirillov conjecture:

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs ⇑ any Ad(Pn) invariant distribution on GLn is Ad(GLn) invariant ⇑ any Ad(GLn−1) invariant distribution on GLn is transposition invariant

Trace formula and relative trace formula: Smooth matching

Other applications of invariant distributions

Study of the principal series

Kirillov conjecture: Any irreducible unitary representation of GLn remains irreducible when restricted to Pn

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs ⇑ any Ad(GLn−1) invariant distribution on GLn is transposition invariant

Trace formula and relative trace formula: Smooth matching

Other applications of invariant distributions

Study of the principal series

Kirillov conjecture: Any irreducible unitary representation of GLn remains irreducible when restricted to Pn ⇑ any Ad(Pn) invariant distribution on GLn is Ad(GLn) invariant

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Trace formula and relative trace formula: Smooth matching

Other applications of invariant distributions

Study of the principal series

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Smooth matching

Other applications of invariant distributions

Study of the principal series

Trace formula and relative trace formula:

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Other applications of invariant distributions

Study of the principal series

Trace formula and relative trace formula: Smooth matching

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Classical examples

Pair Anti-involution (G × G, ∆G) (g, h) 7→ (h−1, g−1) (O(n + k), O(n) × O(k)) (U(n + k), U(n) × U(k)) g 7→ g−1 (GL(n, R), O(n)) g 7→ gt (G, Gθ), where G - Lie group, θ- involution, g 7→ θ(g−1) Gθ is compact (G, K ), where G - is a reductive group, Cartan anti-involution K - maximal compact subgroup

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Results on Gelfand pairs

Pair p-adic case real case (G, (N, ψ)) Gelfand-Kazhdan Shalika, Kostant (GLn(E), GLn(F)) Flicker (GLn+k , GLn × GLk ) Jacquet-Rallis A.- (On+k , On × Ok ) over C Gourevitch (GLn, On) over C (GL2n, Sp2n) Heumos-Rallis A.-Sayag g u (GL , ( , ψ)) Jacquet-Rallis A.-Gourevitch 2n 0 g -Jacquet SP u (GL , ( , ψ)) Offen-Sayag A.-Offen-Sayag n 0 N

real: R and C

p-adic: Qp and its ﬁnite extensions.

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Results on Gelfand pairs Pair p-adic charF > 0 real (G, (N, ψ)) Gelfand- Gelfand- Shalika, Kostant Kazhdan Kazhdan (GLn(E), GLn(F)) Flicker Flicker (GLn+k , GLn × GLk ) Jacquet- A.- Avni- A.- Rallis Gourevitch Gourevitch (On+k , On × Ok ) over C (GLn, On) over C (GL2n, Sp2n) Heumos- Heumos- A.- Rallis Rallis Sayag g u (GL , ( , ψ)) Jacquet- A.-Gourevitch 2n 0 g Rallis -Jacquet SP u (GL , ( , ψ)) Offen-Sayag Offen-Sayag A.-Offen- n 0 N Sayag

real: R and C

p-adic: Qp and its ﬁnite extensions.

charF > 0: Fq((t)) Results on strong Gelfand pairs

Pair p-adic charF > 0 real A.- A.-Avni- A.-Gourevitch, (GLn+1, GLn) Gourevitch- Gourevitch, Sun-Zhu Rallis- Henniart (O(V ⊕ F), O(V )) Schiffmann (U(V ⊕ F), U(V )) Sun-Zhu

real: R and C

p-adic: Qp and its ﬁnite extensions.

charF > 0: Fq((t))

A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs