Bounds on multiplicities of spherical spaces over finite fields

Transcription

1 Bounds on multiplicities of spherical spaces over finite fields A. Aizenbud Weizmann Institute of Science joint with Nir Avni A. Aizenbud Bounds on multiplicities 1 / 9

2 Main conjecture Conjecture Let G be a reductive algebraic group scheme and X be a spherical G space (i.e. over any geometric point of spec(z), the Borel acts with finitely may orbits on X). A. Aizenbud Bounds on multiplicities 2 / 9

3 Main conjecture Conjecture Let G be a reductive algebraic group scheme and X be a spherical G space (i.e. over any geometric point of spec(z), the Borel acts with finitely may orbits on X). Then ( ) sup F is a finite or local field sup dim Hom(S(X(F)), ρ) ρ irr(g(f )) <. A. Aizenbud Bounds on multiplicities 2 / 9

4 previews results sup F is a finite or local field ( sup dim Hom(S(X(F)), ρ) ρ irr(g(f )) ). A. Aizenbud Bounds on multiplicities 3 / 9

5 previews results sup F is a finite or local field ( sup dim Hom(S(X(F)), ρ) ρ irr(g(f )) Delorme, Sakellaridis-Venkatesh finite multiplicity for non-archemedian fields for wide class of spherical spaces. ). A. Aizenbud Bounds on multiplicities 3 / 9

6 previews results sup F is a finite or local field ( sup dim Hom(S(X(F)), ρ) ρ irr(g(f )) Delorme, Sakellaridis-Venkatesh finite multiplicity for non-archemedian fields for wide class of spherical spaces. Kobayashi-Oshima, Krötz-Schlichtkrull bounds on multiplicity for Archemedian fields for wide class of spherical spaces. ). A. Aizenbud Bounds on multiplicities 3 / 9

7 previews results sup F is a finite or local field ( sup dim Hom(S(X(F)), ρ) ρ irr(g(f )) Delorme, Sakellaridis-Venkatesh finite multiplicity for non-archemedian fields for wide class of spherical spaces. Kobayashi-Oshima, Krötz-Schlichtkrull bounds on multiplicity for Archemedian fields for wide class of spherical spaces. Gelfand pairs: ). A. Aizenbud Bounds on multiplicities 3 / 9

8 previews results sup F is a finite or local field ( sup dim Hom(S(X(F)), ρ) ρ irr(g(f )) Delorme, Sakellaridis-Venkatesh finite multiplicity for non-archemedian fields for wide class of spherical spaces. Kobayashi-Oshima, Krötz-Schlichtkrull bounds on multiplicity for Archemedian fields for wide class of spherical spaces. Gelfand pairs: Gelfand-Kazhdan, Shalika, Jacquet-Rallis, A.-Gourevitch-Rallis-Schiffman,... ). A. Aizenbud Bounds on multiplicities 3 / 9

9 previews results sup F is a finite or local field ( sup dim Hom(S(X(F)), ρ) ρ irr(g(f )) Delorme, Sakellaridis-Venkatesh finite multiplicity for non-archemedian fields for wide class of spherical spaces. Kobayashi-Oshima, Krötz-Schlichtkrull bounds on multiplicity for Archemedian fields for wide class of spherical spaces. Gelfand pairs: Gelfand-Kazhdan, Shalika, Jacquet-Rallis, A.-Gourevitch-Rallis-Schiffman,... Cuspidal Gelfand pairs: Hakim,... ). A. Aizenbud Bounds on multiplicities 3 / 9

10 Main result We proved the conjecture if the group is of type A and the fields are finite: A. Aizenbud Bounds on multiplicities 4 / 9

11 Main result We proved the conjecture if the group is of type A and the fields are finite: Theorem (A.-Avni) Let G be a reductive algebraic group scheme of type A and X be a spherical G space. A. Aizenbud Bounds on multiplicities 4 / 9

12 Main result We proved the conjecture if the group is of type A and the fields are finite: Theorem (A.-Avni) Let G be a reductive algebraic group scheme of type A and X be a spherical G space. Then ( ) max dim Hom(ρ, C[X(F)]) <. ρ irr(g(f )) sup F is a finite field A. Aizenbud Bounds on multiplicities 4 / 9

13 Main result We proved the conjecture if the group is of type A and the fields are finite: Theorem (A.-Avni) Let G be a reductive algebraic group scheme of type A and X be a spherical G space. Then ( ) max dim Hom(ρ, C[X(F)]) <. ρ irr(g(f )) sup F is a finite field Idea of the proof: A. Aizenbud Bounds on multiplicities 4 / 9

14 Main result We proved the conjecture if the group is of type A and the fields are finite: Theorem (A.-Avni) Let G be a reductive algebraic group scheme of type A and X be a spherical G space. Then ( ) max dim Hom(ρ, C[X(F)]) <. ρ irr(g(f )) sup F is a finite field Idea of the proof: Use Lusztig s character sheaves in order to categorify the computation of multiplicity of principal series representations. A. Aizenbud Bounds on multiplicities 4 / 9

15 Main result We proved the conjecture if the group is of type A and the fields are finite: Theorem (A.-Avni) Let G be a reductive algebraic group scheme of type A and X be a spherical G space. Then ( ) max dim Hom(ρ, C[X(F)]) <. ρ irr(g(f )) sup F is a finite field Idea of the proof: Use Lusztig s character sheaves in order to categorify the computation of multiplicity of principal series representations. The multiplicities are of geometric nature and lim sup dim Hom(ρ, C[X(F p n)]) is bounded. n A. Aizenbud Bounds on multiplicities 4 / 9

16 Main result We proved the conjecture if the group is of type A and the fields are finite: Theorem (A.-Avni) Let G be a reductive algebraic group scheme of type A and X be a spherical G space. Then ( ) max dim Hom(ρ, C[X(F)]) <. ρ irr(g(f )) sup F is a finite field Idea of the proof: Use Lusztig s character sheaves in order to categorify the computation of multiplicity of principal series representations. The multiplicities are of geometric nature and lim sup dim Hom(ρ, C[X(F p n)]) is bounded. n Deduce the result. A. Aizenbud Bounds on multiplicities 4 / 9

17 Main tool Lusztig s character sheaves Theorem (Lusztig, Shoji) Let G be an algebraic group of type GL defined over F q. For every irreducible representation ρ of G(F q ), there is an induced character sheaf M together with a Weil structure α : Frob qm M which is pure of weight zero, such that χ M,α = χ ρ. A. Aizenbud Bounds on multiplicities 5 / 9

18 Main tool Lusztig s character sheaves Theorem (Lusztig, Shoji) Let G be an algebraic group of type GL defined over F q. For every irreducible representation ρ of G(F q ), there is an induced character sheaf M together with a Weil structure α : Frob qm M which is pure of weight zero, such that χ M,α = χ ρ. G = {B B, g B} π G. A. Aizenbud Bounds on multiplicities 5 / 9

19 Main tool Lusztig s character sheaves Theorem (Lusztig, Shoji) Let G be an algebraic group of type GL defined over F q. For every irreducible representation ρ of G(F q ), there is an induced character sheaf M together with a Weil structure α : Frob qm M which is pure of weight zero, such that χ M,α = χ ρ. G = {B B, g B} π G. M is a (perversed) direct summand of π (K), for some line bundle K on G. A. Aizenbud Bounds on multiplicities 5 / 9

20 Dimension of the orbit space Notation Let algebraic group H act on a variety Y. Denote Y H := {(y, h) Y H hy = y}. A. Aizenbud Bounds on multiplicities 6 / 9

21 Dimension of the orbit space Notation Let algebraic group H act on a variety Y. Denote Y H := {(y, h) Y H hy = y}. Examples A. Aizenbud Bounds on multiplicities 6 / 9

22 Dimension of the orbit space Notation Let algebraic group H act on a variety Y. Denote Y H := {(y, h) Y H hy = y}. Examples If Y is transitive then Y H is smooth and dim Y H = dim H. A. Aizenbud Bounds on multiplicities 6 / 9

23 Dimension of the orbit space Notation Let algebraic group H act on a variety Y. Denote Y H := {(y, h) Y H hy = y}. Examples If Y is transitive then Y H is smooth and dim Y H = dim H. B G = G. A. Aizenbud Bounds on multiplicities 6 / 9

24 Dimension of the orbit space Notation Let algebraic group H act on a variety Y. Denote Y H := {(y, h) Y H hy = y}. Examples If Y is transitive then Y H is smooth and dim Y H = dim H. B G = G. Y has finitely many orbits iff dim Y H = dim H. A. Aizenbud Bounds on multiplicities 6 / 9

25 Dimension of the orbit space Notation Let algebraic group H act on a variety Y. Denote Y H := {(y, h) Y H hy = y}. Examples If Y is transitive then Y H is smooth and dim Y H = dim H. B G = G. Y has finitely many orbits iff dim Y H = dim H. dim(x B) G = dim G iff X is spherical. A. Aizenbud Bounds on multiplicities 6 / 9

26 Categorification of the computation of multiplicity of principal series representations X G π (X G/B) G f p G = (G/B) G f G q pt π A. Aizenbud Bounds on multiplicities 7 / 9

27 Categorification of the computation of multiplicity of principal series representations X G π (X G/B) G f p G = (G/B) G f G q pt π dim Hom(ρ, C[X(F)]) A. Aizenbud Bounds on multiplicities 7 / 9

28 Categorification of the computation of multiplicity of principal series representations X G π (X G/B) G f p G = (G/B) G f G q pt π dim Hom(ρ, C[X(F)]) = χ ρ, χ C[X(F )] A. Aizenbud Bounds on multiplicities 7 / 9

29 Categorification of the computation of multiplicity of principal series representations X G π (X G/B) G f p G = (G/B) G f G q pt π dim Hom(ρ, C[X(F)]) = χ ρ, χ C[X(F )] = χ M, f! (1 XG ) A. Aizenbud Bounds on multiplicities 7 / 9

30 Categorification of the computation of multiplicity of principal series representations X G π (X G/B) G f p G = (G/B) G f G q pt π dim Hom(ρ, C[X(F)]) = χ ρ, χ C[X(F )] = χ M, f! (1 XG ) = χ M, f! (χ CXG ) A. Aizenbud Bounds on multiplicities 7 / 9

31 Categorification of the computation of multiplicity of principal series representations X G π (X G/B) G f p G = (G/B) G f G q pt π dim Hom(ρ, C[X(F)]) = χ ρ, χ C[X(F )] = χ M, f! (1 XG ) = χ M, f! (χ CXG ) = χ M, χ f! (C XG ) A. Aizenbud Bounds on multiplicities 7 / 9

32 Categorification of the computation of multiplicity of principal series representations X G π (X G/B) G f p G = (G/B) G f G q pt π dim Hom(ρ, C[X(F)]) = χ ρ, χ C[X(F )] = χ M, f! (1 XG ) = χ M, f! (χ CXG ) = χ M, χ f! (C XG ) χ π (K), χ f! (C XG ) A. Aizenbud Bounds on multiplicities 7 / 9

33 Categorification of the computation of multiplicity of principal series representations X G π (X G/B) G f p G = (G/B) G f G q pt π dim Hom(ρ, C[X(F)]) = χ ρ, χ C[X(F )] = χ M, f! (1 XG ) = χ M, f! (χ CXG ) = χ M, χ f! (C XG ) χ π (K), χ f! (C XG ) = 1 G(F ) χ q! (π! (K) L f! (C XG )) A. Aizenbud Bounds on multiplicities 7 / 9

34 Categorification of the computation of multiplicity of principal series representations X G π (X G/B) G f p G = (G/B) G f G q pt π dim Hom(ρ, C[X(F)]) = χ ρ, χ C[X(F )] = χ M, f! (1 XG ) = χ M, f! (χ CXG ) = χ M, χ f! (C XG ) χ π (K), χ f! (C XG ) = 1 G(F ) χ = 1 q! (π! (K) f L G(F ) χ! (C XG )) (q p)! ( f (K) π (C XG )) A. Aizenbud Bounds on multiplicities 7 / 9

35 The proof for fixed characteristic Conclusion We constructed a variety Z := (X B) G of dimension dim G such that for any irreducible representation ρ irr(g(f q )), there exist a representation ρ ρ, a line bundle F on Z and wight 0 Weil structure β on H (Z, F) s.t. dim Hom(ρ, C[X(F q )]) = tr(β) G(F q )

36 The proof for fixed characteristic Conclusion We constructed a variety Z := (X B) G of dimension dim G such that for any irreducible representation ρ irr(g(f q )), there exist a representation ρ ρ, a line bundle F on Z and wight 0 Weil structure β on H (Z, F) s.t. Notation M(n) := tr(βn H (Z,F) ) G(F q n ). dim Hom(ρ, C[X(F q )]) = tr(β) G(F q )

37 The proof for fixed characteristic Conclusion We constructed a variety Z := (X B) G of dimension dim G such that for any irreducible representation ρ irr(g(f q )), there exist a representation ρ ρ, a line bundle F on Z and wight 0 Weil structure β on H (Z, F) s.t. Notation M(n) := tr(βn H (Z,F) ) G(F q n ). We have dim Hom(ρ, C[X(F q )]) = tr(β) G(F q )

38 The proof for fixed characteristic Conclusion We constructed a variety Z := (X B) G of dimension dim G such that for any irreducible representation ρ irr(g(f q )), there exist a representation ρ ρ, a line bundle F on Z and wight 0 Weil structure β on H (Z, F) s.t. Notation M(n) := tr(βn H (Z,F) ) G(F q n ). dim Hom(ρ, C[X(F q )]) = We have lim sup M(n) #IrrComp(Z ). n tr(β) G(F q )

39 The proof for fixed characteristic Conclusion We constructed a variety Z := (X B) G of dimension dim G such that for any irreducible representation ρ irr(g(f q )), there exist a representation ρ ρ, a line bundle F on Z and wight 0 Weil structure β on H (Z, F) s.t. Notation M(n) := tr(βn H (Z,F) ) G(F q n ). dim Hom(ρ, C[X(F q )]) = tr(β) G(F q ) We have lim sup M(n) #IrrComp(Z ). n M(n) = Q(v n ), where Q is a rational function on C d and v (C ) d.

40 End of the proof for groups of type GL Lemma Suppose Q is a rational function on C d. Let v (C ) d such that Q is regular at v n, for all n Z >0, and the set {Q(v n ) n Z >0 } is finite. A. Aizenbud Bounds on multiplicities 9 / 9

41 End of the proof for groups of type GL Lemma Suppose Q is a rational function on C d. Let v (C ) d such that Q is regular at v n, for all n Z >0, and the set {Q(v n ) n Z >0 } is finite. Then the function n Q(v n ) is periodic on Z. A. Aizenbud Bounds on multiplicities 9 / 9

42 End of the proof for groups of type GL Lemma Suppose Q is a rational function on C d. Let v (C ) d such that Q is regular at v n, for all n Z >0, and the set {Q(v n ) n Z >0 } is finite. Then the function n Q(v n ) is periodic on Z. dim Hom(ρ, C[X(F q )]) A. Aizenbud Bounds on multiplicities 9 / 9

43 End of the proof for groups of type GL Lemma Suppose Q is a rational function on C d. Let v (C ) d such that Q is regular at v n, for all n Z >0, and the set {Q(v n ) n Z >0 } is finite. Then the function n Q(v n ) is periodic on Z. dim Hom(ρ, C[X(F q )]) M(1) A. Aizenbud Bounds on multiplicities 9 / 9

44 End of the proof for groups of type GL Lemma Suppose Q is a rational function on C d. Let v (C ) d such that Q is regular at v n, for all n Z >0, and the set {Q(v n ) n Z >0 } is finite. Then the function n Q(v n ) is periodic on Z. dim Hom(ρ, C[X(F q )]) M(1) lim sup M(n) n A. Aizenbud Bounds on multiplicities 9 / 9

45 End of the proof for groups of type GL Lemma Suppose Q is a rational function on C d. Let v (C ) d such that Q is regular at v n, for all n Z >0, and the set {Q(v n ) n Z >0 } is finite. Then the function n Q(v n ) is periodic on Z. dim Hom(ρ, C[X(F q )]) M(1) lim sup M(n) #IrrComp(Z ) n A. Aizenbud Bounds on multiplicities 9 / 9