A comparison of Paley-Wiener theorems

Transcription

1 A comparison of Paley-Wiener theorems for groups and symmetric spaces Erik van den Ban Department of Mathematics University of Utrecht International Conference on Integral Geometry, Harmonic Analysis and Representation Theory In honor of Sigurdur Helgason on the occasion of his 80-th birthday Reykjavik, August 18, 2007

2 Outline Paley-Wiener theorems Euclidean space Riemannian symmetric space Spherical Fourier transform Helgason s Paley-Wiener theorems Paley-Wiener theorems for the group Minimal principal series Arthur-Campoli conditions Paley-Wiener theorems for the group Delorme s intertwining conditions Comparison of Paley-Wiener spaces Jets of families of representations Reformulation of conditions The role of the Hecke algebra

3 Euclidean space Setting a : finite dimensional Euclidean space; Fourier transform F : Cc (a) O(a C ), Ff (λ) = f (X)e λ(x) dx; C R (a) := {f C c (a) supp f B(0; R)}. a Definition (Paley-Wiener space) PW R (a) := {ϕ O(a C ) sup(1+ λ )N e R Reλ ϕ(λ) < }. Paley-Wiener theorem F : C R (a) PWR (a) (topol. lin. isom. onto).

4 Euclidean space Setting a : finite dimensional Euclidean space; Fourier transform F : Cc (a) O(a C ), Ff (λ) = f (X)e λ(x) dx; C R (a) := {f C c (a) supp f B(0; R)}. a Definition (Paley-Wiener space) PW R (a) := {ϕ O(a C ) sup(1+ λ )N e R Reλ ϕ(λ) < }. Paley-Wiener theorem F : C R (a) PWR (a) (topol. lin. isom. onto).

5 Euclidean space Setting a : finite dimensional Euclidean space; Fourier transform F : Cc (a) O(a C ), Ff (λ) = f (X)e λ(x) dx; C R (a) := {f C c (a) supp f B(0; R)}. a Definition (Paley-Wiener space) PW R (a) := {ϕ O(a C ) sup(1+ λ )N e R Reλ ϕ(λ) < }. Paley-Wiener theorem F : C R (a) PWR (a) (topol. lin. isom. onto).

6 Non-compact Riemannian symmetric space Setting G semisimple, connected, finite center; K maximal compact; G = KAN Iwasawa decomposition; Σ = Σ(g, a), Σ + positive system; A + positive chamber; W : Weylgroup. Polar decomposition G = K A + K ; C R (K \G/K ) := {f C c (K \G/K ) supp f K exp B(0; R)K };

7 Non-compact Riemannian symmetric space Setting G semisimple, connected, finite center; K maximal compact; G = KAN Iwasawa decomposition; Σ = Σ(g, a), Σ + positive system; A + positive chamber; W : Weylgroup. Polar decomposition G = K A + K ; C R (K \G/K ) := {f C c (K \G/K ) supp f K exp B(0; R)K };

8 Spherical Fourier transform Spherical principal series P = MAN minimal parabolic subgroup π λ := Ind G P (1 λ 1) in Γ L 2(G P C λ+ρ ) K -fixed element: 1 λ (kan) = a λ ρ π λ is realizable in L 2 (K /M) (since G K M P). Definition (Spherical Fourier transform) F : C c (G/K ) K O(a C ) L2 (K /M) K, Ff (λ) = π λ (f )1 λ = G/K f (x) π λ(x)1 λ dx Note F : C c (K \G/K ) O(a C ) C O(a C ).

9 Spherical Fourier transform Spherical principal series P = MAN minimal parabolic subgroup π λ := Ind G P (1 λ 1) in Γ L 2(G P C λ+ρ ) K -fixed element: 1 λ (kan) = a λ ρ π λ is realizable in L 2 (K /M) (since G K M P). Definition (Spherical Fourier transform) F : C c (G/K ) K O(a C ) L2 (K /M) K, Ff (λ) = π λ (f )1 λ = G/K f (x) π λ(x)1 λ dx Note F : C c (K \G/K ) O(a C ) C O(a C ).

10 Spherical Fourier transform Spherical principal series P = MAN minimal parabolic subgroup π λ := Ind G P (1 λ 1) in Γ L 2(G P C λ+ρ ) K -fixed element: 1 λ (kan) = a λ ρ π λ is realizable in L 2 (K /M) (since G K M P). Definition (Spherical Fourier transform) F : C c (G/K ) K O(a C ) L2 (K /M) K, Ff (λ) = π λ (f )1 λ = G/K f (x) π λ(x)1 λ dx Note F : C c (K \G/K ) O(a C ) C O(a C ).

11 Helgason s Paley-Wiener theorems Theorem (Helgason 66, estimates by Gangolli 71) F : C R (K \G/K ) PWR (a) W. Theorem (Helgason 73 ) F : C R (G/K ) K PW R (G/K ). Definition (Paley-Wiener space ) PW R (G/K ) = {ϕ PW R (a) L 2 (K /M) K ϕ(wλ) = A w (λ)ϕ(λ) w W } where A w (λ) End(L 2 (K /M) K ) normalized standard intertwiner (coeffs rational in λ).

12 Helgason s Paley-Wiener theorems Theorem (Helgason 66, estimates by Gangolli 71) F : C R (K \G/K ) PWR (a) W. Theorem (Helgason 73 ) F : C R (G/K ) K PW R (G/K ). Definition (Paley-Wiener space ) PW R (G/K ) = {ϕ PW R (a) L 2 (K /M) K ϕ(wλ) = A w (λ)ϕ(λ) w W } where A w (λ) End(L 2 (K /M) K ) normalized standard intertwiner (coeffs rational in λ).

13 Helgason s Paley-Wiener theorems Theorem (Helgason 66, estimates by Gangolli 71) F : C R (K \G/K ) PWR (a) W. Theorem (Helgason 73 ) F : C R (G/K ) K PW R (G/K ). Definition (Paley-Wiener space ) PW R (G/K ) = {ϕ PW R (a) L 2 (K /M) K ϕ(wλ) = A w (λ)ϕ(λ) w W } where A w (λ) End(L 2 (K /M) K ) normalized standard intertwiner (coeffs rational in λ).

14 The group Minimal principal series P = MAN minimal parabolic subgroup; π ξ,λ = Ind G P (ξ λ 1), for ξ M, λ a C, realizable in V (ξ) := {f L 2 (K ) H ξ f (km) = ξ(m) 1 f (k)}. Definition (Minimal Fourier transform) F : C R (G) K K O(a C ) End( ). where End( ) := [ ξ EndV (ξ) ] K K. Ff (λ) ξ := π ξ,λ (f ) End(V (ξ)) K K for f C c (G) K K, ξ M and λ a C,

15 The group Minimal principal series P = MAN minimal parabolic subgroup; π ξ,λ = Ind G P (ξ λ 1), for ξ M, λ a C, realizable in V (ξ) := {f L 2 (K ) H ξ f (km) = ξ(m) 1 f (k)}. Definition (Minimal Fourier transform) F : C R (G) K K O(a C ) End( ). where End( ) := [ ξ EndV (ξ) ] K K. Ff (λ) ξ := π ξ,λ (f ) End(V (ξ)) K K for f C c (G) K K, ξ M and λ a C,

16 Arthur s Paley-Wiener space Definition (Taylor functionals) O(a C ) tayl := {L : a C S(a) supp L finite}. O(a C ) ϕ Lϕ = λ suppl [L λϕ] (λ). Definition (Arthur-Campoli functionals) AC(G, K ) = {L O(a C ) tayl End( ) K K L = 0 on λ (π ξ,λ (x) ξ M)}. PW space PW Arth (G, K ) = {ϕ PW(a) End( ) Lϕ = 0 L AC(G, K )}.

17 Arthur s Paley-Wiener space Definition (Taylor functionals) O(a C ) tayl := {L : a C S(a) supp L finite}. O(a C ) ϕ Lϕ = λ suppl [L λϕ] (λ). Definition (Arthur-Campoli functionals) AC(G, K ) = {L O(a C ) tayl End( ) K K L = 0 on λ (π ξ,λ (x) ξ M)}. PW space PW Arth (G, K ) = {ϕ PW(a) End( ) Lϕ = 0 L AC(G, K )}.

18 Arthur s Paley-Wiener space Definition (Taylor functionals) O(a C ) tayl := {L : a C S(a) supp L finite}. O(a C ) ϕ Lϕ = λ suppl [L λϕ] (λ). Definition (Arthur-Campoli functionals) AC(G, K ) = {L O(a C ) tayl End( ) K K L = 0 on λ (π ξ,λ (x) ξ M)}. PW space PW Arth (G, K ) = {ϕ PW(a) End( ) Lϕ = 0 L AC(G, K )}.

19 PW theorems for the group Theorem (Arthur 82, Campoli for split rank one) F : C c (G, K ) PW Arth (G, K ). History (incomplete!) Helgason 66, 73: G/K (not a special case); Zhelobenko 74: G complex; Delorme 82: one conjugacy class of Cartan; Arthur 82: general G; Delorme-Clozel 84: trace Paley-Wiener theorem; vdb & Schlichtkrull 03: generalization to Cc (G/H) K ; Delorme 04: general G, different formulation & proof; vdb & Schlichtkrull 05: generalization to E (G/H) K.

20 PW theorems for the group Theorem (Arthur 82, Campoli for split rank one) F : C c (G, K ) PW Arth (G, K ). History (incomplete!) Helgason 66, 73: G/K (not a special case); Zhelobenko 74: G complex; Delorme 82: one conjugacy class of Cartan; Arthur 82: general G; Delorme-Clozel 84: trace Paley-Wiener theorem; vdb & Schlichtkrull 03: generalization to Cc (G/H) K ; Delorme 04: general G, different formulation & proof; vdb & Schlichtkrull 05: generalization to E (G/H) K.

21 PW theorems for the group Theorem (Arthur 82, Campoli for split rank one) F : C c (G, K ) PW Arth (G, K ). History (incomplete!) Helgason 66, 73: G/K (not a special case); Zhelobenko 74: G complex; Delorme 82: one conjugacy class of Cartan; Arthur 82: general G; Delorme-Clozel 84: trace Paley-Wiener theorem; vdb & Schlichtkrull 03: generalization to Cc (G/H) K ; Delorme 04: general G, different formulation & proof; vdb & Schlichtkrull 05: generalization to E (G/H) K.

22 Families of representations Differentiation of family λ π ξ,λ holomorphic family of representations; differentiation in a direction η a gives new holomorphic family π (η) ξ,λ ; Set D of data: each datum D D describes sequences of such differentations followed by taking direct sums; datum D yields π (D) representation in V (D) ; ϕ PW(a) End( ) yields ϕ (D) End(V (D) ).

23 Families of representations Differentiation of family λ π ξ,λ holomorphic family of representations; differentiation in a direction η a gives new holomorphic family π (η) ξ,λ ; Set D of data: each datum D D describes sequences of such differentations followed by taking direct sums; datum D yields π (D) representation in V (D) ; ϕ PW(a) End( ) yields ϕ (D) End(V (D) ).

24 Families of representations Differentiation of family λ π ξ,λ holomorphic family of representations; differentiation in a direction η a gives new holomorphic family π (η) ξ,λ ; Set D of data: each datum D D describes sequences of such differentations followed by taking direct sums; datum D yields π (D) representation in V (D) ; ϕ PW(a) End( ) yields ϕ (D) End(V (D) ).

25 Families of representations Differentiation of family λ π ξ,λ holomorphic family of representations; differentiation in a direction η a gives new holomorphic family π (η) ξ,λ ; Set D of data: each datum D D describes sequences of such differentations followed by taking direct sums; datum D yields π (D) representation in V (D) ; ϕ PW(a) End( ) yields ϕ (D) End(V (D) ).

26 Families of representations Differentiation of family λ π ξ,λ holomorphic family of representations; differentiation in a direction η a gives new holomorphic family π (η) ξ,λ ; Set D of data: each datum D D describes sequences of such differentations followed by taking direct sums; datum D yields π (D) representation in V (D) ; ϕ PW(a) End( ) yields ϕ (D) End(V (D) ).

27 Delorme s PW space Definition (Delorme s PW space) PW Del (G, K ) consists of ϕ PW(a) End( ) satisfying the following intertwining relations. For each datum D D and all submodules U j V j V (D), j = 1, 2, (1) ϕ (D) preserves U 1, V 1 and U 2, V 2 ; (2) for all T Hom g,k (V 1 /U 1, V 2 /U 2 ), the following diagram commutes: V 1 /U 1 V 2 /U 2 ϕ (D) V 1 /U 1 ϕ (D) V 2 /U 2 Theorem (Delorme 05) F : C c (G) K K PWDel (G, K ).

28 Delorme s PW space Definition (Delorme s PW space) PW Del (G, K ) consists of ϕ PW(a) End( ) satisfying the following intertwining relations. For each datum D D and all submodules U j V j V (D), j = 1, 2, (1) ϕ (D) preserves U 1, V 1 and U 2, V 2 ; (2) for all T Hom g,k (V 1 /U 1, V 2 /U 2 ), the following diagram commutes: V 1 /U 1 V 2 /U 2 ϕ (D) V 1 /U 1 ϕ (D) V 2 /U 2 Theorem (Delorme 05) F : C c (G) K K PWDel (G, K ).

29 Delorme s PW space Definition (Delorme s PW space) PW Del (G, K ) consists of ϕ PW(a) End( ) satisfying the following intertwining relations. For each datum D D and all submodules U j V j V (D), j = 1, 2, (1) ϕ (D) preserves U 1, V 1 and U 2, V 2 ; (2) for all T Hom g,k (V 1 /U 1, V 2 /U 2 ), the following diagram commutes: V 1 /U 1 V 2 /U 2 ϕ (D) V 1 /U 1 ϕ (D) V 2 /U 2 Theorem (Delorme 05) F : C c (G) K K PWDel (G, K ).

30 Delorme s PW space Definition (Delorme s PW space) PW Del (G, K ) consists of ϕ PW(a) End( ) satisfying the following intertwining relations. For each datum D D and all submodules U j V j V (D), j = 1, 2, (1) ϕ (D) preserves U 1, V 1 and U 2, V 2 ; (2) for all T Hom g,k (V 1 /U 1, V 2 /U 2 ), the following diagram commutes: V 1 /U 1 V 2 /U 2 ϕ (D) V 1 /U 1 ϕ (D) V 2 /U 2 Theorem (Delorme 05) F : C c (G) K K PWDel (G, K ).

31 Comparison of the Paley-Wiener spaces Natural problem Prove that PW Arth (G, K ) = PW Del (G, K ) without using the Paley-Wiener theorems. Solution: joint work with Sofien Souaifi Involves three key ideas (1) Use jets instead of repeated differentiations 1 ; (2) Reformulate Delorme s intertwining conditions; (3) Use the Hecke algebra (multipl n = convolution) H(G, K ) := E K (G) K K. 1 idea goes back to Casselman

32 Comparison of the Paley-Wiener spaces Natural problem Prove that PW Arth (G, K ) = PW Del (G, K ) without using the Paley-Wiener theorems. Solution: joint work with Sofien Souaifi Involves three key ideas (1) Use jets instead of repeated differentiations 1 ; (2) Reformulate Delorme s intertwining conditions; (3) Use the Hecke algebra (multipl n = convolution) H(G, K ) := E K (G) K K. 1 idea goes back to Casselman

33 Comparison of the Paley-Wiener spaces Natural problem Prove that PW Arth (G, K ) = PW Del (G, K ) without using the Paley-Wiener theorems. Solution: joint work with Sofien Souaifi Involves three key ideas (1) Use jets instead of repeated differentiations 1 ; (2) Reformulate Delorme s intertwining conditions; (3) Use the Hecke algebra (multipl n = convolution) H(G, K ) := E K (G) K K. 1 idea goes back to Casselman

34 Comparison of the Paley-Wiener spaces Natural problem Prove that PW Arth (G, K ) = PW Del (G, K ) without using the Paley-Wiener theorems. Solution: joint work with Sofien Souaifi Involves three key ideas (1) Use jets instead of repeated differentiations 1 ; (2) Reformulate Delorme s intertwining conditions; (3) Use the Hecke algebra (multipl n = convolution) H(G, K ) := E K (G) K K. 1 idea goes back to Casselman

35 Comparison of the Paley-Wiener spaces Natural problem Prove that PW Arth (G, K ) = PW Del (G, K ) without using the Paley-Wiener theorems. Solution: joint work with Sofien Souaifi Involves three key ideas (1) Use jets instead of repeated differentiations 1 ; (2) Reformulate Delorme s intertwining conditions; (3) Use the Hecke algebra (multipl n = convolution) H(G, K ) := E K (G) K K. 1 idea goes back to Casselman

36 Jets of families of representations Jet space O 0 : space of germs at 0 of holomorphic functions on a C ; m 0 O 0 : unique maximal ideal; I m 0 : ideal such that k : m k 0 I m 0. Jet space: O 0 /I (finite dimensional /C). Jets of families O(a C ) ϕ ϕ(i) O(a C ) O 0/I. O(a C ) Hom(V, W ) φ ϕ (I) O(a C ) Hom(O 0/I V, O 0 /I W ). π ξ,λ, V (ξ) π (I) ξ,λ, O 0/I V (ξ).

37 Jets of families of representations Jet space O 0 : space of germs at 0 of holomorphic functions on a C ; m 0 O 0 : unique maximal ideal; I m 0 : ideal such that k : m k 0 I m 0. Jet space: O 0 /I (finite dimensional /C). Jets of families O(a C ) ϕ ϕ(i) O(a C ) O 0/I. O(a C ) Hom(V, W ) φ ϕ (I) O(a C ) Hom(O 0/I V, O 0 /I W ). π ξ,λ, V (ξ) π (I) ξ,λ, O 0/I V (ξ).

38 Jets of families of representations Jet space O 0 : space of germs at 0 of holomorphic functions on a C ; m 0 O 0 : unique maximal ideal; I m 0 : ideal such that k : m k 0 I m 0. Jet space: O 0 /I (finite dimensional /C). Jets of families O(a C ) ϕ ϕ(i) O(a C ) O 0/I. O(a C ) Hom(V, W ) φ ϕ (I) O(a C ) Hom(O 0/I V, O 0 /I W ). π ξ,λ, V (ξ) π (I) ξ,λ, O 0/I V (ξ).

39 Jets of families of representations Jet space O 0 : space of germs at 0 of holomorphic functions on a C ; m 0 O 0 : unique maximal ideal; I m 0 : ideal such that k : m k 0 I m 0. Jet space: O 0 /I (finite dimensional /C). Jets of families O(a C ) ϕ ϕ(i) O(a C ) O 0/I. O(a C ) Hom(V, W ) φ ϕ (I) O(a C ) Hom(O 0/I V, O 0 /I W ). π ξ,λ, V (ξ) π (I) ξ,λ, O 0/I V (ξ).

40 Reformulation of Delorme s intertwining conditions Direct sums of jets Datum: I O 0, ξ 1,..., ξ n M, µ 1,..., µ n a C ; π (I) ξ,µ := j π (I) ξ j,µ j in j V (ξ j ) (I) ; PW(a) End( ) ϕ ϕ (I) ξ,µ j End(V (ξ j ) (I) ); Definition For V a Harish-Chandra module: End(V ) # := {ϕ End(V ) K K n U < V n : ϕ n (U) U}. Delorme s condition: ϕ PW(a) End( ) belongs to PW Del (G, K ) if and only if for all data (I, ξ, µ) : ϕ (I) ξ,µ End(V (I) ξ,µ )#.

41 Reformulation of Delorme s intertwining conditions Direct sums of jets Datum: I O 0, ξ 1,..., ξ n M, µ 1,..., µ n a C ; π (I) ξ,µ := j π (I) ξ j,µ j in j V (ξ j ) (I) ; PW(a) End( ) ϕ ϕ (I) ξ,µ j End(V (ξ j ) (I) ); Definition For V a Harish-Chandra module: End(V ) # := {ϕ End(V ) K K n U < V n : ϕ n (U) U}. Delorme s condition: ϕ PW(a) End( ) belongs to PW Del (G, K ) if and only if for all data (I, ξ, µ) : ϕ (I) ξ,µ End(V (I) ξ,µ )#.

42 Reformulation of Delorme s intertwining conditions Direct sums of jets Datum: I O 0, ξ 1,..., ξ n M, µ 1,..., µ n a C ; π (I) ξ,µ := j π (I) ξ j,µ j in j V (ξ j ) (I) ; PW(a) End( ) ϕ ϕ (I) ξ,µ j End(V (ξ j ) (I) ); Definition For V a Harish-Chandra module: End(V ) # := {ϕ End(V ) K K n U < V n : ϕ n (U) U}. Delorme s condition: ϕ PW(a) End( ) belongs to PW Del (G, K ) if and only if for all data (I, ξ, µ) : ϕ (I) ξ,µ End(V (I) ξ,µ )#.

43 Reformulation of Delorme s intertwining conditions Direct sums of jets Datum: I O 0, ξ 1,..., ξ n M, µ 1,..., µ n a C ; π (I) ξ,µ := j π (I) ξ j,µ j in j V (ξ j ) (I) ; PW(a) End( ) ϕ ϕ (I) ξ,µ j End(V (ξ j ) (I) ); Definition For V a Harish-Chandra module: End(V ) # := {ϕ End(V ) K K n U < V n : ϕ n (U) U}. Delorme s condition: ϕ PW(a) End( ) belongs to PW Del (G, K ) if and only if for all data (I, ξ, µ) : ϕ (I) ξ,µ End(V (I) ξ,µ )#.

44 Reformulation of Delorme s intertwining conditions Direct sums of jets Datum: I O 0, ξ 1,..., ξ n M, µ 1,..., µ n a C ; π (I) ξ,µ := j π (I) ξ j,µ j in j V (ξ j ) (I) ; PW(a) End( ) ϕ ϕ (I) ξ,µ j End(V (ξ j ) (I) ); Definition For V a Harish-Chandra module: End(V ) # := {ϕ End(V ) K K n U < V n : ϕ n (U) U}. Delorme s condition: ϕ PW(a) End( ) belongs to PW Del (G, K ) if and only if for all data (I, ξ, µ) : ϕ (I) ξ,µ End(V (I) ξ,µ )#.

45 Reformulation of Arthur-Campoli conditions Definition For (π, V ) a Harish-Chandra module: π(g) = {ψ End(V ) K K g G : ψ, π(g) = 0}. Lemma (Arthur-Campoli conditions) ϕ PW(a) End( ) belongs to PW Arth (G, K ) if and only if for all data (I, ξ, µ), ϕ (I) ξ,µ π(i) ξ,λ (G). Delorme s condition: ϕ PW(a) End( ) belongs to PW Del (G, K ) if and only if for all data (I, ξ, µ) : ϕ (I) ξ,µ End(V (I) ξ,µ )#.

46 Reformulation of Arthur-Campoli conditions Definition For (π, V ) a Harish-Chandra module: π(g) = {ψ End(V ) K K g G : ψ, π(g) = 0}. Lemma (Arthur-Campoli conditions) ϕ PW(a) End( ) belongs to PW Arth (G, K ) if and only if for all data (I, ξ, µ), ϕ (I) ξ,µ π(i) ξ,λ (G). Delorme s condition: ϕ PW(a) End( ) belongs to PW Del (G, K ) if and only if for all data (I, ξ, µ) : ϕ (I) ξ,µ End(V (I) ξ,µ )#.

47 Reformulation of Arthur-Campoli conditions Definition For (π, V ) a Harish-Chandra module: π(g) = {ψ End(V ) K K g G : ψ, π(g) = 0}. Lemma (Arthur-Campoli conditions) ϕ PW(a) End( ) belongs to PW Arth (G, K ) if and only if for all data (I, ξ, µ), ϕ (I) ξ,µ π(i) ξ,λ (G). Delorme s condition: ϕ PW(a) End( ) belongs to PW Del (G, K ) if and only if for all data (I, ξ, µ) : ϕ (I) ξ,µ End(V (I) ξ,µ )#.

48 The role of the Hecke algebra Recall Category of Harish-Chandra modules category of finite modules for H(G, K ) (= E K (G) K K U(g) U(k) E (K ) K K ); Theorem (vdb - Souaifi) Let (π, V ) be a Harish-Chandra module. Then End(V ) # = π(h(g, K )); π(g) = π(h(g, K )). Corollary PW Del (G, K ) = PW Arth (G, K ). Proof. For all data (I, ξ, µ), ϕ (I) ξ,µ End( j V (ξ j ) (I) ) # ϕ (I) ξ,µ π(i) ξ,µ (G).

49 The role of the Hecke algebra Recall Category of Harish-Chandra modules category of finite modules for H(G, K ) (= E K (G) K K U(g) U(k) E (K ) K K ); Theorem (vdb - Souaifi) Let (π, V ) be a Harish-Chandra module. Then End(V ) # = π(h(g, K )); π(g) = π(h(g, K )). Corollary PW Del (G, K ) = PW Arth (G, K ). Proof. For all data (I, ξ, µ), ϕ (I) ξ,µ End( j V (ξ j ) (I) ) # ϕ (I) ξ,µ π(i) ξ,µ (G).

50 The role of the Hecke algebra Recall Category of Harish-Chandra modules category of finite modules for H(G, K ) (= E K (G) K K U(g) U(k) E (K ) K K ); Theorem (vdb - Souaifi) Let (π, V ) be a Harish-Chandra module. Then End(V ) # = π(h(g, K )); π(g) = π(h(g, K )). Corollary PW Del (G, K ) = PW Arth (G, K ). Proof. For all data (I, ξ, µ), ϕ (I) ξ,µ End( j V (ξ j ) (I) ) # ϕ (I) ξ,µ π(i) ξ,µ (G).

51 The role of the Hecke algebra Recall Category of Harish-Chandra modules category of finite modules for H(G, K ) (= E K (G) K K U(g) U(k) E (K ) K K ); Theorem (vdb - Souaifi) Let (π, V ) be a Harish-Chandra module. Then End(V ) # = π(h(g, K )); π(g) = π(h(g, K )). Corollary PW Del (G, K ) = PW Arth (G, K ). Proof. For all data (I, ξ, µ), ϕ (I) ξ,µ End( j V (ξ j ) (I) ) # ϕ (I) ξ,µ π(i) ξ,µ (G).

52 The role of the Hecke algebra Recall Category of Harish-Chandra modules category of finite modules for H(G, K ) (= E K (G) K K U(g) U(k) E (K ) K K ); Theorem (vdb - Souaifi) Let (π, V ) be a Harish-Chandra module. Then End(V ) # = π(h(g, K )); π(g) = π(h(g, K )). Corollary PW Del (G, K ) = PW Arth (G, K ). Proof. For all data (I, ξ, µ), ϕ (I) ξ,µ End( j V (ξ j ) (I) ) # ϕ (I) ξ,µ π(i) ξ,µ (G).

53 Comparison with Helgason s theorems F(H(G, K )) P(a ) End( ) subalgebra; Theorem (Reformulation of Arthur s) ϕ PW(a) End( ) belongs to PW Arth (G, K ) iff for all choices λ 1,..., λ n a C, u 1,..., u n S(a ), and ξ 1,..., ξ n M there exists a p F(H(G, K )) such that u j ϕ ξj (λ j ) = u j p ξj (λ j ) ( 1 j n). Lemma (a) F(pr 1 H(G, K ) δ K ) = P(a ) W. (b) F(H(G, K ) δ K ) = {ϕ P(a ) L 2 (K /M) K w W : ϕ(wλ) = A w (λ)ϕ(λ)}. Proof of (b) uses Helgason s thm. Arthur s thm + Lemma = Helgason s thm.

54 Comparison with Helgason s theorems F(H(G, K )) P(a ) End( ) subalgebra; Theorem (Reformulation of Arthur s) ϕ PW(a) End( ) belongs to PW Arth (G, K ) iff for all choices λ 1,..., λ n a C, u 1,..., u n S(a ), and ξ 1,..., ξ n M there exists a p F(H(G, K )) such that u j ϕ ξj (λ j ) = u j p ξj (λ j ) ( 1 j n). Lemma (a) F(pr 1 H(G, K ) δ K ) = P(a ) W. (b) F(H(G, K ) δ K ) = {ϕ P(a ) L 2 (K /M) K w W : ϕ(wλ) = A w (λ)ϕ(λ)}. Proof of (b) uses Helgason s thm. Arthur s thm + Lemma = Helgason s thm.

55 Comparison with Helgason s theorems F(H(G, K )) P(a ) End( ) subalgebra; Theorem (Reformulation of Arthur s) ϕ PW(a) End( ) belongs to PW Arth (G, K ) iff for all choices λ 1,..., λ n a C, u 1,..., u n S(a ), and ξ 1,..., ξ n M there exists a p F(H(G, K )) such that u j ϕ ξj (λ j ) = u j p ξj (λ j ) ( 1 j n). Lemma (a) F(pr 1 H(G, K ) δ K ) = P(a ) W. (b) F(H(G, K ) δ K ) = {ϕ P(a ) L 2 (K /M) K w W : ϕ(wλ) = A w (λ)ϕ(λ)}. Proof of (b) uses Helgason s thm. Arthur s thm + Lemma = Helgason s thm.

56 Comparison with Helgason s theorems F(H(G, K )) P(a ) End( ) subalgebra; Theorem (Reformulation of Arthur s) ϕ PW(a) End( ) belongs to PW Arth (G, K ) iff for all choices λ 1,..., λ n a C, u 1,..., u n S(a ), and ξ 1,..., ξ n M there exists a p F(H(G, K )) such that u j ϕ ξj (λ j ) = u j p ξj (λ j ) ( 1 j n). Lemma (a) F(pr 1 H(G, K ) δ K ) = P(a ) W. (b) F(H(G, K ) δ K ) = {ϕ P(a ) L 2 (K /M) K w W : ϕ(wλ) = A w (λ)ϕ(λ)}. Proof of (b) uses Helgason s thm. Arthur s thm + Lemma = Helgason s thm.

57 Dear Siggi Thanks for inspiring us with your beautiful math

58 Happy Birthday!

59 Ann. Math. 1973

60 The Paley-Wiener theorem