Bounds on multiplicities of spherical spaces over finite fields A. - PowerPoint PPT Presentation

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

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

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 sup dim Hom ( S ( X ( F )) , ρ ) < ∞ . F is a finite or local field ρ ∈ irr ( G ( F )) A. Aizenbud Bounds on multiplicities 2 / 9

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

previews results � � sup sup dim Hom ( S ( X ( F )) , ρ ) . F is a finite or local field 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

previews results � � sup sup dim Hom ( S ( X ( F )) , ρ ) . F is a finite or local field 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

previews results � � sup sup dim Hom ( S ( X ( F )) , ρ ) . F is a finite or local field 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

previews results � � sup sup dim Hom ( S ( X ( F )) , ρ ) . F is a finite or local field 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

previews results � � sup sup dim Hom ( S ( X ( F )) , ρ ) . F is a finite or local field 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

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

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

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 � � sup ρ ∈ irr ( G ( F )) dim Hom ( ρ, C [ X ( F )]) max < ∞ . F is a finite field A. Aizenbud Bounds on multiplicities 4 / 9

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 � � sup ρ ∈ irr ( G ( F )) dim Hom ( ρ, C [ X ( F )]) max < ∞ . F is a finite field Idea of the proof: A. Aizenbud Bounds on multiplicities 4 / 9

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 � � sup ρ ∈ irr ( G ( F )) dim Hom ( ρ, C [ X ( F )]) max < ∞ . 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

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 � � sup ρ ∈ irr ( G ( F )) dim Hom ( ρ, C [ X ( F )]) max < ∞ . 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

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 � � sup ρ ∈ irr ( G ( F )) dim Hom ( ρ, C [ X ( F )]) max < ∞ . 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

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 ∗ q M → M which is pure of weight zero, such that χ M ,α = χ ρ . A. Aizenbud Bounds on multiplicities 5 / 9

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 ∗ q M → 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

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 ∗ q M → 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

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

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

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

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

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

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

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

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