Paley-Wiener theorem(s) for real reductive Lie groups

Paley-Wiener theorem(s) for real reductive Lie groups

Sofiane Souaifi¹

1Institut de Recherche Mathématique Avancée Université de Strasbourg

PVs- Tokyo, September 1-5, 2014

joint work with E. P. van den Ban

Outline

1 Introduction

2 The Paley-Wiener theorem(s) forG

3 Holomorphic families of representations and their successive derivatives

4 The generalized Hecke algebra for(G, K)

5 The Paley-Wiener theorem(s) forGand some reformulation

6 Application

Outline

1 Introduction

2 The Paley-Wiener theorem(s) forG

3 Holomorphic families of representations and their successive derivatives

4 The generalized Hecke algebra for(G, K)

5 The Paley-Wiener theorem(s) forGand some reformulation

6 Application

Paley-Wiener theorem?

Describe the image of the Fourier transform

LetGbe a real reductive Lie group of the Harish-Chandra class, e.g.Gsemisimple, connected, with finite center.

LetK=G^θ,θthe associated Cartan involution.

Letg:=Lie(G)andg₌k_⊕p, its Cartan decomposition w.r.t.θ^. Letabe a fixed maximal abelian subspace ofp, A=exp (a).

For instance, ifG=SL(n,R),then one can takeK=SO(n),a_'Rⁿ⁻¹_.

The classical Euclidean case

Consider the Euclidean spacea,a_C:=a_⊗C,a^∗_C dual ofa_C.

Paley-Wiener thm

(C_c^∞(a))^∧=PW(a).

with PW(a) :={ϕ∈O(a^∗_C) : ∃R>0∀n>0∃C_n>0,

|ϕ(λ)| ≤Cn(1+ |λ|)⁻ⁿe^R|Reλ|}

Fourier transformof f ∈C_c^∞(a): fb(λ) :=R

af(X)π1,λ(X)dX,λ∈a^∗_C whereπ1,λ1-dim. rep-n ofa(as an additive group):π1,λ(X)=e^−λ(X)

The classical Euclidean case

Consider the Euclidean spacea,a_C:=a_⊗C,a^∗_C dual ofa_C.

Paley-Wiener thm

(C^∞_c (a)R)^∧=PW(a)R. with C^∞_c (a)R:={f ∈C^∞_c (a) : ksupp(f)k ≤R}, and

PW(a_)R:={ϕ∈O(a^∗_{C) : ∀n>0∃Cn>0,}

|ϕ(λ)| ≤Cn(1+ |λ|)⁻ⁿe^R|Reλ|}

Fourier transformof f ∈C_c^∞(a): fb(λ) :=R

af(X)π1,λ(X)dX,λ∈a^∗_C whereπ1,λ1-dim. rep-n ofa(as an additive group):π1,λ(X)=e^−λ(X)

Outline

1 Introduction

2 The Paley-Wiener theorem(s) forG

3 Holomorphic families of representations and their successive derivatives

4 The generalized Hecke algebra for(G, K)

5 The Paley-Wiener theorem(s) forGand some reformulation

6 Application

The (operator valued) Fourier transform

LetPbe a fixed minimal parabolic subgroup, withP=MANits Langlands decomposition,

e.g., ifG=SL(n,R),Pis the upper triangular group withAblock-diagonal,N strictly upper triangular.

LetM^∧be the unitary dual ofM.

Definition (Minimal principal series ofG)

For(ξ,λ)∈M^∧×a^∗_C,πξ,λis the right regular rep-n ofG^onC^∞(G :ξ⊗λ)^where:

C^∞(G :ξ⊗λ) :={ψ∈C^∞(G, V_ξ) :ψ(manx)=a^λ+ρPξ(m)ψ(x)}

The (operator valued) Fourier transform

LetPbe a fixed minimal parabolic subgroup, withP=MANits Langlands decomposition,

e.g., ifG=SL(n,R),Pis the upper triangular group withAblock-diagonal,N strictly upper triangular.

LetM^∧be the unitary dual ofM.

Definition (Minimal principal series ofG)

For(ξ,λ)∈M^∧×a^∗_C,πξ,λis the rep-n of the compact realization of the smooth minimal p-s ofGonC^∞(K :ξ), given bytransport of structure fromπξ,λunder res-n toK :

C^∞(G :ξ⊗λ)−→^' C^∞(K :ξ)

Example

ConsiderG=SL(2,R).HereMb ={+,−}, N={³1 y 0 1

´}anda^∗_C'C.Under restriction of functions onGtoθ(N),the rep-nπ±,λis equiv-t top_±,λ

p_±,λ³a b c d

´ f(y)=







|b y+d|^1+λf³_{a y+c}

b y+d

´

if+ sgn(b y+d)|b y+d|^1+λf³_{a y+c}

b y+d

´ if−.

Definition (Fourier transform) For f ∈C_c^∞(G),ξ∈M^∧,

fb(ξ,λ) :=Z

G

f(x)π_ξ,λ(x)d x, λ∈a^∗_C. f 7→fbmapsC^∞_c (G)into⊕_ξ∈M^∧O(a^∗_C)⊗End(C^∞(K :ξ)).

Example

ConsiderG=SL(2,R).HereMb ={+,−}, N={³1 y 0 1

´}anda^∗_C'C.Under restriction of functions onGtoθ(N),the rep-nπ±,λis equiv-t top_±,λ

p_±,λ³a b c d

´ f(y)=







|b y+d|^1+λf³_{a y+c}

b y+d

´

if+ sgn(b y+d)|b y+d|^1+λf³_{a y+c}

b y+d

´ if−.

Definition (Fourier transform) For f ∈C_c^∞(G),ξ∈M^∧,

fb(ξ,λ) :=Z

G

f(x)π_ξ,λ(x)d x, λ∈a^∗_C. f 7→fbmapsC^∞_c (G)^into⊕_ξ∈M^∧O(a^∗_C)⊗End(C^∞_{(K :ξ)).}

The Paley-Wiener theorems

LetC^∞_c (K\G/K)denote the subspace of bi-K-invariant elements inC_c^∞(G).

LetWthe Weyl group associated to(g,a).

Theorem (Helgason ’66, estimates by Gangolli ’71) C_c^∞(K\G/K)^∧=PW(a₎^W

LetC^∞_c (G/K)Kdenote the subspace of rightK-invariant and leftK-finite elements inC_c^∞(G).

Theorem (Helgason ’73)

(C^∞_c (G/K)K)^∧=PW(G/K)

with PW(G/K)={ϕ∈PW(a)⊗L²(K/M) :ϕ(wλ)=Aw(λ)ϕ(λ)} whereAw(λ)∈End(L²(K/M)K)normalized standard intertwining operator (with rational coeff-s inλ^).

The Paley-Wiener theorems

LetC^∞_c (K\G/K)denote the subspace of bi-K-invariant elements inC_c^∞(G).

LetWthe Weyl group associated to(g,a).

Theorem (Helgason ’66, estimates by Gangolli ’71) C_c^∞(K\G/K)^∧=PW(a₎^W

LetC^∞_c (G/K)Kdenote the subspace of rightK-invariant and leftK-finite elements inC_c^∞(G).

Theorem (Helgason ’73)

(C^∞_c (G/K)K)^∧=PW(G/K)

with PW(G/K)={ϕ∈PW(a)⊗L²(K/M) :ϕ(wλ)=Aw(λ)ϕ(λ)}

whereAw(λ)∈End(L²(K/M)K)normalized standard intertwining operator (with rational coeff-s inλ^).

LetC^∞_c (G, K)denote the space of bi-K-finite elements inC^∞_c (G).

Forξ∈M^{∧, let}

S(ξ) :=End(C^∞(K :ξ)_K×K).

Theorem (Arthur ’82, Campoli ’80 for real rank one case) C_c^∞(G, K)^∧=PWArth(G, K)

with PWArth(G, K)={ϕ∈PW(a)⊗(⊕ξ∈M^∧S(ξ)) : ϕsatisfiesA-C}.

Theorem (Delorme ’06)

C^∞_c (G, K)^∧=PWDel(G, K)

with PWDel(G, K)={ϕ∈PW(a)⊗(⊕_ξ∈M^∧S(ξ)) : ϕsatisfiesintert. rel-ns}.

Without usingthe Paley-Wiener theorems, we want to show PWArth(G, K)=PWDel(G, K)

LetC^∞_c (G, K)denote the space of bi-K-finite elements inC^∞_c (G).

Forξ∈M^{∧, let}

S(ξ) :=End(C^∞(K :ξ)_K×K).

Theorem (Arthur ’82, Campoli ’80 for real rank one case) C_c^∞(G, K)^∧=PWArth(G, K)

with PW_Arth(G, K)={ϕ∈PW(a)⊗(⊕ξ∈M^∧S(ξ)) :ϕsatisfiesA-C}.

Theorem (Delorme ’06)

C^∞_c (G, K)^∧=PWDel(G, K)

with PWDel(G, K)={ϕ∈PW(a)⊗(⊕_ξ∈M^∧S(ξ)) : ϕsatisfiesintert. rel-ns}.

Without usingthe Paley-Wiener theorems, we want to show PWArth(G, K)=PWDel(G, K)

LetC^∞_c (G, K)denote the space of bi-K-finite elements inC^∞_c (G).

Forξ∈M^{∧, let}

S(ξ) :=End(C^∞(K :ξ)_K×K).

Theorem (Arthur ’82, Campoli ’80 for real rank one case) C_c^∞(G, K)^∧=PWArth(G, K)

with PW_Arth(G, K)={ϕ∈PW(a)⊗(⊕ξ∈M^∧S(ξ)) :ϕsatisfiesA-C}.

Theorem (Delorme ’06)

C^∞_c (G, K)^∧=PWDel(G, K)

with PWDel(G, K)={ϕ∈PW(a)⊗(⊕_ξ∈M^∧S(ξ)) : ϕsatisfiesintert. rel-ns}.

Without usingthe Paley-Wiener theorems, we want to show PWArth(G, K)=PWDel(G, K)

LetC^∞_c (G, K)denote the space of bi-K-finite elements inC^∞_c (G).

Forξ∈M^{∧, let}

S(ξ) :=End(C^∞(K :ξ)_K×K).

Theorem (Arthur ’82, Campoli ’80 for real rank one case) C_c^∞(G, K)^∧=PWArth(G, K)

with PW_Arth(G, K)={ϕ∈PW(a)⊗(⊕ξ∈M^∧S(ξ)) :ϕsatisfiesA-C}.

Theorem (Delorme ’06)

C^∞_c (G, K)^∧=PWDel(G, K)

with PWDel(G, K)={ϕ∈PW(a)⊗(⊕_ξ∈M^∧S(ξ)) : ϕsatisfiesintert. rel-ns}.

Without usingthe Paley-Wiener theorems, we want to show PWArth(G, K)=PWDel(G, K)

Some notation

For anyη∈a^∗_C,Φ∈O(a^∗_C), let define Φ(λ;η) := d

d z

¡Φ(λ+zη)¢

|z=0, λ∈a^∗_C. The map

η7→[Φ7→Φ(·;η)]

extends uniquely to an algebra hom ofS(a^∗_C).

For anyη∈a^∗_C andΦ∈O(a^∗_C, End(V))(Va F-space), let define Φ^(η)∈O(a^∗_C, End(V⊕V))by

Φ^(η)(λ) :=³_{Φ(λ) Φ(λ;η)}

0 Φ(λ)

´ .

By iterating the process, one can generalize this definition to any finite sequence ina^∗_C.

Arthur-Campoli relations

An Arthur-Campoli sequence is a finite sequence(ξi,ψi,λi,ui)i withξi∈M^∧, ψ∈S(ξi)^∗_K×K,λi∈a^∗_C andui∈S(a^∗_C)s.t

X

i

〈πξi,λi;u_i(x),ψi〉 =0, x∈G.

Let

F:= ⊕_ξ∈M^∧O(a^∗_C)⊗S(ξ). Definition

ϕ∈F satisfies theArthur-Campoli relations if X

i

〈ϕ(ξi,λi;ui),ψi〉 =0, for any (ξi,ψi,λi,ui)iA-C sequence.

Arthur-Campoli relations

An Arthur-Campoli sequence is a finite sequence(ξi,ψi,λi,ui)i withξi∈M^∧, ψ∈S(ξi)^∗_K×K,λi∈a^∗_C andui∈S(a^∗_C)s.t

X

i

〈πξi,λi;u_i(x),ψi〉 =0, x∈G.

Let

F:= ⊕_ξ∈M^∧O(a^∗_C)⊗S(ξ).

Definition

ϕ∈F satisfies theArthur-Campoli relations if X

i

〈ϕ(ξi,λi;ui),ψi〉 =0, for any (ξi,ψi,λi,ui)iA-C sequence.

Intertwining relations

Letδ=(ξ,λ,η)∈D^withξ∈M^∧,λ∈a^∗_C andηa finite sequence ina^∗_C. Define the rep-nπδofGinV_πδ:=C^∞(K :ξ)^(η)byπδ:=π⁽_ξ^η_,λ⁾.

Similarly, define forϕ∈F,ϕδ∈End(C^∞(K :ξ)^(η))byϕδ:=ϕ^(η)(ξ,λ).

Definition (Intertwining relations)

A functionϕ∈F satisfies Delorme’sintertwining relationsif

for everyN∈Z+and eachδ∈D^N,ϕ_δpreserves all closed inv-nt s-spaces ofπδ;

for allN1, N2∈Z+,allδ1∈D^{N1 and}δ2∈D^N2,and any two sequences of closed invariant subspacesUj⊂Vjforπδj,

V1/U1 V2/U2

V1/U1 V2/U2

ϕ_δ1

T T

ϕ_δ2

The space of functionsϕ∈F satisfying (a) and (b) is denoted byF(D).

Intertwining relations

Letδ=(ξ,λ,η)∈D^withξ∈M^∧,λ∈a^∗_C andηa finite sequence ina^∗_C. Define the rep-nπδofGinV_πδ:=C^∞(K :ξ)^(η)byπδ:=π⁽_ξ^η_,λ⁾.

Similarly, define forϕ∈F,ϕδ∈End(C^∞(K :ξ)^(η))byϕδ:=ϕ^(η)(ξ,λ).

Definition (Intertwining relations)

A functionϕ∈F satisfies Delorme’sintertwining relationsif

for everyN∈Z+and eachδ∈D^N,ϕ_δpreserves all closed inv-nt s-spaces ofπδ;

for allN1, N2∈Z+,allδ1∈D^{N1 and}δ2∈D^N2,and any two sequences of closed invariant subspacesUj⊂Vjforπδj,

V1/U1 V2/U2

V1/U1 V2/U2

ϕ_δ1

T T

ϕ_δ2

The space of functionsϕ∈F satisfying (a) and (b) is denoted byF(D).

Some examples of rank one

where “non-classical” conditions occur already.

Example (PW forG=SU(2, 1)on2-dim-lK-types)

Besides growth and symmetric cond-ns, one has extra cond-ns related to holomorphic families of intertwining op-rs between p. s. rep-ns.

Example (PW forG=SU(2, 1)on4-dim-lK-types)

Besides growth and symmetric cond-ns, one has extra cond-ns related to derivatives ofholomorphic families of intertwining op-rs between p. s. rep-ns.

Some examples of rank one

where “non-classical” conditions occur already.

Example (PW forG=SU(2, 1)on2-dim-lK-types)

Besides growth and symmetric cond-ns, one has extra cond-ns related to holomorphic families of intertwining op-rs between p. s. rep-ns.

Example (PW forG=SU(2, 1)on4-dim-lK-types)

Besides growth and symmetric cond-ns, one has extra cond-ns related to derivatives ofholomorphic families of intertwining op-rs between p. s. rep-ns.

Paley-Wiener thm:C_c^∞(G, K)^∧=PW(G, K)

besides growth cond-ns

Arthur-Campoli relations on some families of rep-ns

Arthur

Intertwining relations on some families of rep-ns

Delor me

Hecke algebraH(G, K)

Relations in terms ofH(G, K) π(G)^⊥⊥=End(V)^#

π(G)^⊥⊥=π(H(G, K)) π(H(G, K))=End(V)^#

Paley-Wiener thm:C_c^∞(G, K)^∧=PW(G, K)

besides growth cond-ns

Arthur-Campoli relations on some families of rep-ns

Arthur

Intertwining relations on some families of rep-ns

Delor me

Hecke algebraH(G, K)

Relations in terms ofH(G, K) π(G)^⊥⊥=End(V)^#

π(G)^⊥⊥=π(H(G, K)) π(H(G, K))=End(V)^#

Paley-Wiener thm:C_c^∞(G, K)^∧=PW(G, K)

besides growth cond-ns

Arthur-Campoli relations on some families of rep-ns

Arthur

Intertwining relations on some families of rep-ns

Delor me

Hecke algebraH(G, K)

Relations in terms ofH(G, K) π(G)^⊥⊥=End(V)^#

π(G)^⊥⊥=π(H(G, K)) π(H(G, K))=End(V)^#

Paley-Wiener thm:C_c^∞(G, K)^∧=PW(G, K)

besides growth cond-ns

Arthur-Campoli relations on some families of rep-ns

Arthur

Intertwining relations on some families of rep-ns

Delor me

Hecke algebraH(G, K)

Relations in terms ofH(G, K)

π(G)^⊥⊥=End(V)^#

π(G)^⊥⊥=π(H(G, K)) π(H(G, K))=End(V)^#

Paley-Wiener thm:C_c^∞(G, K)^∧=PW(G, K)

besides growth cond-ns

Arthur-Campoli relations on some families of rep-ns

Arthur

Intertwining relations on some families of rep-ns

Delor me

Hecke algebraH(G, K)

Relations in terms ofH(G, K) π(G)^⊥⊥=End(V)^#

π(G)^⊥⊥=π(H(G, K)) π(H(G, K))=End(V)^#

Outline

1 Introduction

2 The Paley-Wiener theorem(s) forG

3 Holomorphic families of representations and their successive derivatives

4 The generalized Hecke algebra for(G, K)

5 The Paley-Wiener theorem(s) forGand some reformulation

6 Application

Holomorphic families of representations

Definition (Holomorphic families of representations)

The pair(π, V)is a holomorphic family of smooth rep-ns ofGovera^∗_C if Vis a Fréchet space

π: G→O(a^∗_{C, End(V))}is a group homomorphism, continuous, with rest-n toKconstant ona^∗_C, and

g7→π(g)(λ)vis C^∞, for all (λ,v)∈a^∗_C×V.

g7→π(g)(λ)v^smooth.

Example (Minimal principal series ofG)

∀ξ∈M^∧, (π_ξ,λ)_λis a holomorphic family of smooth rep-ns ofG.

Holomorphic families of representations

Definition (Holomorphic families of representations)

The pair(π, V)is a holomorphic family of smooth rep-ns ofGovera^∗_C if Vis a Fréchet space

π: G→O(a^∗_C, End(V))is a group homomorphism, continuous, with rest-n toKconstant ona^∗_C, and g7→π(g)(λ)vsmooth.

Example (Minimal principal series ofG)

∀ξ∈M^∧, (πξ,λ)_λis a holomorphic family of smooth rep-ns ofG.

Tensoring with finite dimensional O

-module

LetO0be the algebra of germs at0of holomorphic functions defined on a neighborhood of0ina^∗_C andM its maximal ideal.

Definition (Successive derivatives)

For any finite dimensionalO0-moduleE,^anyΦ∈O(a^∗_C),define Φ^(E)∈O(a^∗_C, End(E))by:

∀_λ∈a^∗_C Φ^(E)(λ) := the germ at 0 ofΦ(· +λ) E

Using canonical identification, we can extend this definition to hol. f-ies of rep-ns (π, V)as follows:(π^(E), V^(E))is the hol. f-y of rep-ns given by:

V^(E):=E⊗V

π^(E)(g,λ) :=(π(g))^(E)(λ), g∈G,λ∈a^∗_C.

Tensoring with finite dimensional O

-module

LetO0be the algebra of germs at0of holomorphic functions defined on a neighborhood of0ina^∗_C andM its maximal ideal.

Definition (Successive derivatives)

For any finite dimensionalO0-moduleE,^anyΦ∈O(a^∗_C),define Φ^(E)∈O(a^∗_C, End(E))by:

∀_λ∈a^∗_C Φ^(E)(λ) := the germ at 0 ofΦ(· +λ) E

Using canonical identification, we can extend this definition to hol. f-ies of rep-ns (π, V)as follows:(π^(E), V^(E))is the hol. f-y of rep-ns given by:

V^(E):=E⊗V

π^(E)(g,λ) :=(π(g))^(E)(λ), g∈G,λ∈a^∗_C.

Tensoring with finite dimensional O

-module

LetO0be the algebra of germs at0of holomorphic functions defined on a neighborhood of0ina^∗_C andM its maximal ideal.

Definition (Successive derivatives)

For any finite dimensionalO0-moduleE,^anyΦ∈O(a^∗_C),define Φ^(E)∈O(a^∗_C, End(E))by:

∀_λ∈a^∗_C Φ^(E)(λ) := the germ at 0 ofΦ(· +λ) E

Using canonical identification, we can extend this definition to hol. f-ies of rep-ns (π, V)as follows:(π^(E), V^(E))is the hol. f-y of rep-ns given by:

V^(E):=E⊗V

π^(E)(g,λ) :=(π(g))^(E)(λ), g∈G,λ∈a^∗_C.

Some examples

For anyu∈S(a^∗_C),one can canonically associate a f-dimO0-moduleEand µ∈End(E)^∗s.t.

Φ(λ;u)=µ◦Φ^(E)(λ), Φ∈O(a^∗_C),λ∈a^∗_C.

For anyη∈a^∗_C, letIη:={ϕ∈M:∂(η)ϕ∈M}.It is a cofinite ideal ofO0. Then

Φ^(η):=³_Φ(

·) Φ(·;η) 0 Φ(·)

´

=Φ^(O0/Iη).

Outline

1 Introduction

2 The Paley-Wiener theorem(s) forG

3 Holomorphic families of representations and their successive derivatives

4 The generalized Hecke algebra for(G, K)

5 The Paley-Wiener theorem(s) forGand some reformulation

6 Application

Definition and general facts

LetH(G, K)be the (convolution) algebra of leftK-finite distributions onG^with support contained inK.

Some properties

H(G, K)is an associative algebra and has an approximation of identity. H(G, K)'U(g_C)⊗U(k_C)H(K)(as linear spaces), given by:

∂(u)∗T 7→ u⊗T

equivariant for leftU(g_C)-action and rightH(K)^-action. H(G, K)'H(K)⊗U(k_C)U(g_C).

All elements ofH(G, K)are bi-K-finite.

Definition and general facts

LetH(G, K)be the (convolution) algebra of leftK-finite distributions onG^with support contained inK.

Some properties

H(G, K)is an associative algebra and has an approximation of identity.

H(G, K)'U(g_C)⊗U(k_C)H(K)(as linear spaces), given by:

∂(u)∗T 7→ u⊗T

equivariant for leftU(g_C)-action and rightH(K)^-action.

H(G, K)'H(K)⊗U(k_C)U(g_C).

All elements ofH(G, K)are bi-K-finite.

H (G, K) -modules and Harish-Chandra modules

A Harish-Chandra moduleVis a(g, K)-module

Example (Underlying(

g

Then, for any f. d.O0-modulesE, V_ξ,λ^(E)is a Harish-Chandra module.

Property

HC-mod ' f. g. adm. appr. unitalH(G, K)-mod

H (G, K) -modules and Harish-Chandra modules

A Harish-Chandra moduleVis a(g, K)-module, i.e Vis a complex vector space endowed with

compatibleU(g_C)andK-module structures: ∀v∈V,k∈K,u∈U(g_C), X∈k, k·(u·v)=Ad(k)u·k·v, _{d t}^d ¡

exp(tX)·v¢

|t=0=X·v), all elements areK-finite

Example (Underlying(

g

Then, for any f. d.O0-modulesE, V_ξ^(E)_,λ is a Harish-Chandra module.

Property

HC-mod ' f. g. adm. appr. unitalH(G, K)-mod

H (G, K) -modules and Harish-Chandra modules

A Harish-Chandra moduleVis a(g, K)-module, s.t.

Vis a finitely generatedU(g_C)-module

Vis admissible (each irreducibleK-module occurs only finitely often inV).

Example (Underlying(

g

Then, for any f. d.O0-modulesE, V_ξ^(E)_,λ is a Harish-Chandra module.

Property

HC-mod ' f. g. adm. appr. unitalH(G, K)-mod

H (G, K) -modules and Harish-Chandra modules

A Harish-Chandra moduleVis a(g, K)-module, s.t.

Vis a finitely generatedU(g_C)-module

Vis admissible (each irreducibleK-module occurs only finitely often inV).

Example (Underlying(

g

Then, for any f. d.O0-modulesE, V_ξ^(E)_,λ is a Harish-Chandra module.

Property

HC-mod ' f. g. adm. appr. unitalH(G, K)-mod

H (G, K) -modules and Harish-Chandra modules

A Harish-Chandra moduleVis a(g, K)-module, s.t.

Vis a finitely generatedU(g_C)-module

Vis admissible (each irreducibleK-module occurs only finitely often inV).

Example (Underlying(

g

Then, for any f. d.O0-modulesE, V_ξ^(E)_,λ is a Harish-Chandra module.

Property

HC-mod ' f. g. adm. appr. unitalH(G, K)-mod

A bi-commutant theorem

Letπbe an admissible Fréchet representation ofGwith finite composition series andVits Harish-Chandra module.

Definition

π(G)^⊥:={ψ∈End(V)^∗_K×K: ∀g∈G,〈ψ,π(g)〉 =0} π(G)^⊥⊥:={ϕ∈End(V)_K×K:∀_ψ∈π(G)^⊥,〈ϕ,ψ〉 =0} End(V)^#:={ϕ∈End(V)_K×K: ∀n∀U<V^⊕n,ϕ^⊕n(U)⊂U}.

Theorem (van den Ban & S.)

Letπ^andVas above. Then, as linear subspaces ofEnd(V)_K×K, End(V)^#=π(H(G, K))

π(G)^⊥⊥=π(H(G, K)).

A bi-commutant theorem

Letπbe an admissible Fréchet representation ofGwith finite composition series andVits Harish-Chandra module.

Definition

π(G)^⊥:={ψ∈End(V)^∗_K×K: ∀g∈G,〈ψ,π(g)〉 =0}

π(G)^⊥⊥:={ϕ∈End(V)_K×K:∀_ψ∈π(G)^⊥,〈ϕ,ψ〉 =0}

End(V)^#:={ϕ∈End(V)_K×K: ∀n∀U<V^⊕n,ϕ^⊕n(U)⊂U}.

Theorem (van den Ban & S.)

Letπ^andVas above. Then, as linear subspaces ofEnd(V)_K×K, End(V)^#=π(H(G, K))

π(G)^⊥⊥=π(H(G, K)).

A bi-commutant theorem

Letπbe an admissible Fréchet representation ofGwith finite composition series andVits Harish-Chandra module.

Definition

π(G)^⊥:={ψ∈End(V)^∗_K×K: ∀g∈G,〈ψ,π(g)〉 =0}

π(G)^⊥⊥:={ϕ∈End(V)_K×K:∀_ψ∈π(G)^⊥,〈ϕ,ψ〉 =0}

End(V)^#:={ϕ∈End(V)_K×K: ∀n∀U<V^⊕n,ϕ^⊕n(U)⊂U}.

Theorem (van den Ban & S.)

Letπ^andVas above. Then, as linear subspaces ofEnd(V)_K×K, End(V)^#=π(H(G, K))

π(G)^⊥⊥=π(H(G, K)).

Outline

1 Introduction

2 The Paley-Wiener theorem(s) forG

3 Holomorphic families of representations and their successive derivatives

4 The generalized Hecke algebra for(G, K)

5 The Paley-Wiener theorem(s) forGand some reformulation

6 Application

The Arthur Paley-Wiener theorem

C_c^∞(G, K)^∧=PWArth(G, K) besides growth cond-ns

Arthur-Campoli relations Arthur

Reformulation π(G)^⊥⊥

Theorem (Arthur ’82)

The Fourier transform is a topological isomorphism fromC^∞_c (G, K)onto PWArth(G, K)with

PW_Arth(G, K)={ϕ∈PW(a)⊗(⊕_ξ∈M^∧S(ξ)) : ϕsatisfies A-C}.

Lemma (“Algebraic” reformulation) For anyϕ∈F,

ϕsatisfies A-C

⇐⇒ ∀(E,ξ,λ)∈O0−modf d×M^∧×a^∗_C,ϕ^(E)(ξ,λ)∈π^(E)_ξ,λ(G)^⊥⊥.

Theorem (Arthur ’82)

The Fourier transform is a topological isomorphism fromC^∞_c (G, K)onto PWArth(G, K)with

PW_Arth(G, K)={ϕ∈PW(a)⊗(⊕_ξ∈M^∧S(ξ)) : ϕsatisfies A-C}.

Lemma (“Algebraic” reformulation) For anyϕ∈F,

ϕsatisfies A-C

⇐⇒ ∀(E,ξ,λ)∈O0−modf d×M^∧×a^∗_C,ϕ^(E)(ξ,λ)∈π^(E)_ξ,λ(G)^⊥⊥.

The Delorme Paley-Wiener theorem

C^∞_c (G, K)^∧=PWDel(G, K) besides growth cond-ns

Intertwining relations Delor

me

Reformulation End(V)^#

Theorem (Delorme ’06)

The Fourier transform is a topological isomorphism fromC^∞_c (G, K)onto PW_Del(G, K)with

PWDel(G, K)={ϕ∈PW(a)⊗(⊕ξ∈M^∧S(ξ)) : ϕ∈F(D)}.

Such a description was first obtained by Zelobenko ’75, in the complex case.

Delorme uses all parabolic subgroups ofG,and not only a minimal one, for his description.

Lemma (“Algebraic” reformulation) For anyϕ∈F,we have:

ϕ∈F(D)

⇐⇒ ∀(E,ξ,λ)∈O0−mod_{f d}×M^∧×a^∗_C,ϕ^(E)(ξ,λ)∈End(V^(E)_ξ,λ)^#.

Theorem (Delorme ’06)

The Fourier transform is a topological isomorphism fromC^∞_c (G, K)onto PW_Del(G, K)with

PWDel(G, K)={ϕ∈PW(a)⊗(⊕ξ∈M^∧S(ξ)) : ϕ∈F(D)}.

Such a description was first obtained by Zelobenko ’75, in the complex case.

Delorme uses all parabolic subgroups ofG,and not only a minimal one, for his description.

Lemma (“Algebraic” reformulation) For anyϕ∈F,we have:

ϕ∈F(D)

⇐⇒ ∀(E,ξ,λ)∈O0−mod_{f d}×M^∧×a^∗_C,ϕ^(E)(ξ,λ)∈End(V^(E)_ξ,λ)^#.

Theorem (Delorme ’06)

The Fourier transform is a topological isomorphism fromC^∞_c (G, K)onto PW_Del(G, K)with

PWDel(G, K)={ϕ∈PW(a)⊗(⊕ξ∈M^∧S(ξ)) : ϕ∈F(D)}.

Such a description was first obtained by Zelobenko ’75, in the complex case.

Delorme uses all parabolic subgroups ofG,and not only a minimal one, for his description.

Lemma (“Algebraic” reformulation) For anyϕ∈F,we have:

ϕ∈F(D)

⇐⇒ ∀(E,ξ,λ)∈O0−modf d×M^∧×a^∗_C,ϕ^(E)(ξ,λ)∈End(V^(E)_ξ,λ)^#.

Outline

1 Introduction

2 The Paley-Wiener theorem(s) forG

3 Holomorphic families of representations and their successive derivatives

4 The generalized Hecke algebra for(G, K)

5 The Paley-Wiener theorem(s) forGand some reformulation

6 Application

Application

Recall that

ϕsatisfies A-C

⇐⇒ ∀(E,ξ,λ)∈O0−modf d×M^∧×a^∗_C,ϕ^(E)(ξ,λ)∈π^(E)_ξ,λ(G)^⊥⊥.

and ϕ∈F(D)

⇐⇒ ∀(E,ξ,λ)∈O0−modf d×M^∧×a^∗_C,ϕ^(E)(ξ,λ)∈End(V_ξ,λ^(E))^#.

Then, as a corollary of

End(V_ξ,λ^(E))^#=π^(E)_ξ,λ(H(G, K))=π^(E)_ξ,λ(G)^⊥⊥, we get

Corollary

PWDel(G, K)=PWArth(G, K).

Application

Recall that

ϕsatisfies A-C

⇐⇒ ∀(E,ξ,λ)∈O0−modf d×M^∧×a^∗_C,ϕ^(E)(ξ,λ)∈π^(E)_ξ,λ(G)^⊥⊥.

and ϕ∈F(D)

⇐⇒ ∀(E,ξ,λ)∈O0−modf d×M^∧×a^∗_C,ϕ^(E)(ξ,λ)∈End(V_ξ,λ^(E))^#. Then, as a corollary of

End(V_ξ,λ^(E))^#=π^(E)_ξ,λ(H(G, K))=π^(E)_ξ,λ(G)^⊥⊥, we get

Corollary

PWDel(G, K)=PWArth(G, K).

Comparison with Helgason’s theorem

Remark that

H(G, K)^∧⊂P(a^∗_{C)⊗End(⊕ξ∈M}∧C^∞(K :ξ)K×K) as a subalgebra.

Theorem (Reformulation of Arthur’s theorem) Letϕ∈ ⊕_ξ∈M^∧PW(a_)⊗S(ξ).One has

ϕ∈PWArth(G, K)

⇐⇒ ∀(ui,ξi,λi)⊂S(a^∗_C)×M^∧×a^∗_C,∃p∈H(G,K)^∧,∀i ϕ(ξi,λi;ui)=p(ξi,λi;ui).

Lemma

LetδK∈H(G, K)denote the distribution onGgiven by1Kd k.Then (pr₁H(G, K)∗δK)^∧=P(a^∗_C)^W

(H(G, K)∗δK)^∧={ϕ∈P(a^∗_C)⊗L²(K/M)K:∀w∈W,ϕ(wλ)=Aw(λ)ϕ(λ)}

Comparison with Helgason’s theorem

Remark that

H(G, K)^∧⊂P(a^∗_{C)⊗End(⊕ξ∈M}∧C^∞(K :ξ)K×K) as a subalgebra.

Theorem (Reformulation of Arthur’s theorem) Letϕ∈ ⊕_ξ∈M^∧PW(a)⊗S(ξ).One has

ϕ∈PWArth(G, K)

⇐⇒ ∀(ui,ξi,λi)⊂S(a^∗_C)×M^∧×a^∗_C,∃p∈H(G,K)^∧,∀i ϕ(ξi,λi;ui)=p(ξi,λi;ui).

Lemma

LetδK∈H(G, K)denote the distribution onGgiven by1Kd k.Then (pr₁H(G, K)∗δK)^∧=P(a^∗_C)^W

(H(G, K)∗δK)^∧={ϕ∈P(a^∗_C)⊗L²(K/M)K:∀w∈W,ϕ(wλ)=Aw(λ)ϕ(λ)}

Comparison with Helgason’s theorem

Remark that

H(G, K)^∧⊂P(a^∗_{C)⊗End(⊕ξ∈M}∧C^∞(K :ξ)K×K) as a subalgebra.

Theorem (Reformulation of Arthur’s theorem) Letϕ∈ ⊕_ξ∈M^∧PW(a)⊗S(ξ).One has

ϕ∈PWArth(G, K)

⇐⇒ ∀(ui,ξi,λi)⊂S(a^∗_C)×M^∧×a^∗_C,∃p∈H(G,K)^∧,∀i ϕ(ξi,λi;ui)=p(ξi,λi;ui).

Lemma

LetδK∈H(G, K)denote the distribution onGgiven by1Kd k.Then (pr₁H(G, K)∗δK)^∧=P(a^∗_C)^W

(H(G, K)∗δK)^∧={ϕ∈P(a^∗_C)⊗L²(K/M)K:∀w∈W,ϕ(wλ)=Aw(λ)ϕ(λ)}

Open problem

In 2007, van den Ban & Schlichtkrull proved a Paley-Wiener theorem for reductive symmetric spaces of the non-compact type. Their description uses some kind of Arthur-Campoli relations.

The Fourier transforms are meromorphic functions.

Question

Give another description in terms of intertwining cond-ns. What will play the role of the Hecke algebra?

Open problem

In 2007, van den Ban & Schlichtkrull proved a Paley-Wiener theorem for reductive symmetric spaces of the non-compact type. Their description uses some kind of Arthur-Campoli relations.

The Fourier transforms are meromorphic functions.

Question

Give another description in terms of intertwining cond-ns.

What will play the role of the Hecke algebra?

Open problem

In 2007, van den Ban & Schlichtkrull proved a Paley-Wiener theorem for reductive symmetric spaces of the non-compact type. Their description uses some kind of Arthur-Campoli relations.

The Fourier transforms are meromorphic functions.

Question

Give another description in terms of intertwining cond-ns. What will play the role of the Hecke algebra?

Open problem

In 2007, van den Ban & Schlichtkrull proved a Paley-Wiener theorem for reductive symmetric spaces of the non-compact type. Their description uses some kind of Arthur-Campoli relations.

The Fourier transforms are meromorphic functions.

Question

Give another description in terms of intertwining cond-ns.

What will play the role of the Hecke algebra?