SLIDE 1 Bounds on multiplicities of spherical spaces
- ver finite fields
- A. Aizenbud
Weizmann Institute of Science
joint with Nir Avni http://aizenbud.org
Bounds on multiplicities 1 / 9
SLIDE 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).
Bounds on multiplicities 2 / 9
SLIDE 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
ρ∈irr(G(F))
dim Hom(S(X(F)), ρ)
Bounds on multiplicities 2 / 9
SLIDE 4 previews results
sup
F is a finite or local field
ρ∈irr(G(F))
dim Hom(S(X(F)), ρ)
Bounds on multiplicities 3 / 9
SLIDE 5 previews results
sup
F is a finite or local field
ρ∈irr(G(F))
dim Hom(S(X(F)), ρ)
Delorme, Sakellaridis-Venkatesh – finite multiplicity for non-Archemedian fields for wide class of spherical spaces.
Bounds on multiplicities 3 / 9
SLIDE 6 previews results
sup
F is a finite or local field
ρ∈irr(G(F))
dim Hom(S(X(F)), ρ)
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.
Bounds on multiplicities 3 / 9
SLIDE 7 previews results
sup
F is a finite or local field
ρ∈irr(G(F))
dim Hom(S(X(F)), ρ)
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:
Bounds on multiplicities 3 / 9
SLIDE 8 previews results
sup
F is a finite or local field
ρ∈irr(G(F))
dim Hom(S(X(F)), ρ)
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,...
Bounds on multiplicities 3 / 9
SLIDE 9 previews results
sup
F is a finite or local field
ρ∈irr(G(F))
dim Hom(S(X(F)), ρ)
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,...
Bounds on multiplicities 3 / 9
SLIDE 10 Main result
We proved the conjecture if the group is of type A and the fields are finite:
Bounds on multiplicities 4 / 9
SLIDE 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.
Bounds on multiplicities 4 / 9
SLIDE 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 sup
F is a finite field
ρ∈irr(G(F)) dim Hom(ρ, C[X(F)])
Bounds on multiplicities 4 / 9
SLIDE 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 sup
F is a finite field
ρ∈irr(G(F)) dim Hom(ρ, C[X(F)])
Idea of the proof:
Bounds on multiplicities 4 / 9
SLIDE 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 sup
F is a finite field
ρ∈irr(G(F)) dim Hom(ρ, C[X(F)])
Idea of the proof: Use Lusztig’s character sheaves in order to categorify the computation of multiplicity of principal series representations.
Bounds on multiplicities 4 / 9
SLIDE 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 sup
F is a finite field
ρ∈irr(G(F)) dim Hom(ρ, C[X(F)])
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
n→∞
dim Hom(ρ, C[X(Fpn)]) is bounded.
Bounds on multiplicities 4 / 9
SLIDE 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 sup
F is a finite field
ρ∈irr(G(F)) dim Hom(ρ, C[X(F)])
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
n→∞
dim Hom(ρ, C[X(Fpn)]) is bounded. Deduce the result.
Bounds on multiplicities 4 / 9
SLIDE 17 Main tool – Lusztig’s character sheaves
Theorem (Lusztig, Shoji) Let G be an algebraic group of type GL defined over Fq. For every irreducible representation ρ of G(Fq), there is an induced character sheaf M together with a Weil structure α : Frob∗
qM → M which is pure of weight zero, such that
χM,α = χρ.
Bounds on multiplicities 5 / 9
SLIDE 18 Main tool – Lusztig’s character sheaves
Theorem (Lusztig, Shoji) Let G be an algebraic group of type GL defined over Fq. For every irreducible representation ρ of G(Fq), there is an induced character sheaf M together with a Weil structure α : Frob∗
qM → M which is pure of weight zero, such that
χM,α = χρ. ˜ G = {B ∈ B, g ∈ B} π → G.
Bounds on multiplicities 5 / 9
SLIDE 19 Main tool – Lusztig’s character sheaves
Theorem (Lusztig, Shoji) Let G be an algebraic group of type GL defined over Fq. For every irreducible representation ρ of G(Fq), there is an induced character sheaf M together with a Weil structure α : Frob∗
qM → M which is pure of weight zero, such that
χM,α = χρ. ˜ G = {B ∈ B, g ∈ B} π → G. M is a (perversed) direct summand of π∗(K), for some line bundle K on ˜ G.
Bounds on multiplicities 5 / 9
SLIDE 20 Dimension of the orbit space
Notation Let algebraic group H act on a variety Y. Denote YH := {(y, h) ∈ Y × H|hy = y}.
Bounds on multiplicities 6 / 9
SLIDE 21 Dimension of the orbit space
Notation Let algebraic group H act on a variety Y. Denote YH := {(y, h) ∈ Y × H|hy = y}. Examples
Bounds on multiplicities 6 / 9
SLIDE 22 Dimension of the orbit space
Notation Let algebraic group H act on a variety Y. Denote YH := {(y, h) ∈ Y × H|hy = y}. Examples If Y is transitive then YH is smooth and dim YH = dim H.
Bounds on multiplicities 6 / 9
SLIDE 23 Dimension of the orbit space
Notation Let algebraic group H act on a variety Y. Denote YH := {(y, h) ∈ Y × H|hy = y}. Examples If Y is transitive then YH is smooth and dim YH = dim H. BG = ˜ G.
Bounds on multiplicities 6 / 9
SLIDE 24 Dimension of the orbit space
Notation Let algebraic group H act on a variety Y. Denote YH := {(y, h) ∈ Y × H|hy = y}. Examples If Y is transitive then YH is smooth and dim YH = dim H. BG = ˜ G. Y has finitely many orbits iff dim YH = dim H.
Bounds on multiplicities 6 / 9
SLIDE 25 Dimension of the orbit space
Notation Let algebraic group H act on a variety Y. Denote YH := {(y, h) ∈ Y × H|hy = y}. Examples If Y is transitive then YH is smooth and dim YH = dim H. BG = ˜ G. Y has finitely many orbits iff dim YH = dim H. dim(X × B)G = dim G iff X is spherical.
Bounds on multiplicities 6 / 9
SLIDE 26 Categorification of the computation of multiplicity of principal series representations
(X × G/B)G
f
G = (G/B)G
π
q
Bounds on multiplicities 7 / 9
SLIDE 27 Categorification of the computation of multiplicity of principal series representations
(X × G/B)G
f
G = (G/B)G
π
q
dim Hom(ρ, C[X(F)])
Bounds on multiplicities 7 / 9
SLIDE 28 Categorification of the computation of multiplicity of principal series representations
(X × G/B)G
f
G = (G/B)G
π
q
dim Hom(ρ, C[X(F)]) = χρ, χC[X(F)]
Bounds on multiplicities 7 / 9
SLIDE 29 Categorification of the computation of multiplicity of principal series representations
(X × G/B)G
f
G = (G/B)G
π
q
dim Hom(ρ, C[X(F)]) = χρ, χC[X(F)] = χM, f!(1XG)
Bounds on multiplicities 7 / 9
SLIDE 30 Categorification of the computation of multiplicity of principal series representations
(X × G/B)G
f
G = (G/B)G
π
q
dim Hom(ρ, C[X(F)]) = χρ, χC[X(F)] = χM, f!(1XG) = χM, f!(χCXG)
Bounds on multiplicities 7 / 9
SLIDE 31 Categorification of the computation of multiplicity of principal series representations
(X × G/B)G
f
G = (G/B)G
π
q
dim Hom(ρ, C[X(F)]) = χρ, χC[X(F)] = χM, f!(1XG) = χM, f!(χCXG) = χM, χf!(CXG)
Bounds on multiplicities 7 / 9
SLIDE 32 Categorification of the computation of multiplicity of principal series representations
(X × G/B)G
f
G = (G/B)G
π
q
dim Hom(ρ, C[X(F)]) = χρ, χC[X(F)] = χM, f!(1XG) = χM, f!(χCXG) = χM, χf!(CXG) ≤ χπ∗(K), χf!(CXG)
Bounds on multiplicities 7 / 9
SLIDE 33 Categorification of the computation of multiplicity of principal series representations
(X × G/B)G
f
G = (G/B)G
π
q
dim Hom(ρ, C[X(F)]) = χρ, χC[X(F)] = χM, f!(1XG) = χM, f!(χCXG) = χM, χf!(CXG) ≤ χπ∗(K), χf!(CXG) =
1 |G(F)|χ q!(π!(K)
L
⊗f!(CXG))
Bounds on multiplicities 7 / 9
SLIDE 34 Categorification of the computation of multiplicity of principal series representations
(X × G/B)G
f
G = (G/B)G
π
q
dim Hom(ρ, C[X(F)]) = χρ, χC[X(F)] = χM, f!(1XG) = χM, f!(χCXG) = χM, χf!(CXG) ≤ χπ∗(K), χf!(CXG) =
1 |G(F)|χ q!(π!(K)
L
⊗f!(CXG)) = 1 |G(F)|χ(q◦p)!(˜ f ∗(K)⊗˜ π∗(CXG))
Bounds on multiplicities 7 / 9
SLIDE 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(Fq)), 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(Fq)]) = tr(β) |G(Fq)|
SLIDE 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(Fq)), 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(Fq)]) = tr(β) |G(Fq)| Notation M(n) :=
tr(βn|H∗(Z,F)) |G(Fqn)|
.
SLIDE 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(Fq)), 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(Fq)]) = tr(β) |G(Fq)| Notation M(n) :=
tr(βn|H∗(Z,F)) |G(Fqn)|
. We have
SLIDE 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(Fq)), 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(Fq)]) = tr(β) |G(Fq)| Notation M(n) :=
tr(βn|H∗(Z,F)) |G(Fqn)|
. We have lim sup
n→∞
M(n) ≤ #IrrComp(Z).
SLIDE 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(Fq)), 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(Fq)]) = tr(β) |G(Fq)| Notation M(n) :=
tr(βn|H∗(Z,F)) |G(Fqn)|
. We have lim sup
n→∞
M(n) ≤ #IrrComp(Z). M(n) = Q(vn), where Q is a rational function on Cd and v ∈ (C×)d.
SLIDE 40 End of the proof for groups of type GL
Lemma Suppose Q is a rational function on Cd. Let v ∈ (C×)d such that Q is regular at vn, for all n ∈ Z>0, and the set {Q(vn)|n ∈ Z>0} is finite.
Bounds on multiplicities 9 / 9
SLIDE 41 End of the proof for groups of type GL
Lemma Suppose Q is a rational function on Cd. Let v ∈ (C×)d such that Q is regular at vn, for all n ∈ Z>0, and the set {Q(vn)|n ∈ Z>0} is finite. Then the function n → Q(vn) is periodic on Z.
Bounds on multiplicities 9 / 9
SLIDE 42 End of the proof for groups of type GL
Lemma Suppose Q is a rational function on Cd. Let v ∈ (C×)d such that Q is regular at vn, for all n ∈ Z>0, and the set {Q(vn)|n ∈ Z>0} is finite. Then the function n → Q(vn) is periodic on Z. dim Hom(ρ, C[X(Fq)])
Bounds on multiplicities 9 / 9
SLIDE 43 End of the proof for groups of type GL
Lemma Suppose Q is a rational function on Cd. Let v ∈ (C×)d such that Q is regular at vn, for all n ∈ Z>0, and the set {Q(vn)|n ∈ Z>0} is finite. Then the function n → Q(vn) is periodic on Z. dim Hom(ρ, C[X(Fq)]) ≤ M(1)
Bounds on multiplicities 9 / 9
SLIDE 44 End of the proof for groups of type GL
Lemma Suppose Q is a rational function on Cd. Let v ∈ (C×)d such that Q is regular at vn, for all n ∈ Z>0, and the set {Q(vn)|n ∈ Z>0} is finite. Then the function n → Q(vn) is periodic on Z. dim Hom(ρ, C[X(Fq)]) ≤ M(1) ≤ lim sup
n→∞
M(n)
Bounds on multiplicities 9 / 9
SLIDE 45 End of the proof for groups of type GL
Lemma Suppose Q is a rational function on Cd. Let v ∈ (C×)d such that Q is regular at vn, for all n ∈ Z>0, and the set {Q(vn)|n ∈ Z>0} is finite. Then the function n → Q(vn) is periodic on Z. dim Hom(ρ, C[X(Fq)]) ≤ M(1) ≤ lim sup
n→∞
M(n) ≤ #IrrComp(Z)
Bounds on multiplicities 9 / 9
