Categorical Weil Representation & Sign Problem
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1 Categorical Weil Representation & Sign Problem Shamgar Gurevich Madison May 16, 2012 Shamgar Gurevich (Madison) Categorical Weil Representation May 16, / 12
2 Joint work with: Ronny Hadani (Math, Austin) Shamgar Gurevich (Madison) Categorical Weil Representation May 16, / 12
3 (0) Motivation - CANONICAL CATEGORY Theorem (Canonical vector space, G Hadani 04 ) There exists a natural functor H : Symp Vect }{{}. over k=f q over C Shamgar Gurevich (Madison) Categorical Weil Representation May 16, / 12
4 (0) Motivation - CANONICAL CATEGORY Theorem (Canonical vector space, G Hadani 04 ) There exists a natural functor For V Symp we have H : Symp Vect }{{}. over k=f q over C ρ V : Sp(V ) GL(H(V )) Weil representation. Shamgar Gurevich (Madison) Categorical Weil Representation May 16, / 12
5 Motivation - CANONICAL CATEGORY Want: lax 2-functor Symp V C(V) canonical category of l-adic sheaves. }{{} In Var over k Shamgar Gurevich (Madison) Categorical Weil Representation May 16, / 12
6 Motivation - CANONICAL CATEGORY Want: lax 2-functor Symp V C(V) canonical category of l-adic sheaves. }{{} In Var over k For V Symp get ρ V : Sp(V) Aut(C(V)) categorical Weil representation. Shamgar Gurevich (Madison) Categorical Weil Representation May 16, / 12
7 (I) Canonical Vector Space - CONSTRUCTION Heisenberg group H = V k, (v, z) (v, z ) = (v + v, z + z ω(v, v )). Shamgar Gurevich (Madison) Categorical Weil Representation May 16, / 12
8 (I) Canonical Vector Space - CONSTRUCTION Heisenberg group H = V k, (v, z) (v, z ) = (v + v, z + z ω(v, v )). Additive character 1 = ψ : Z (H) = k C. Shamgar Gurevich (Madison) Categorical Weil Representation May 16, / 12
9 (I) Canonical Vector Space - CONSTRUCTION Heisenberg group H = V k, (v, z) (v, z ) = (v + v, z + z ω(v, v )). Additive character 1 = ψ : Z (H) = k C. Oriented Lagrangians OLag = {L = (L, o L ); L Lag(V ), o L L 0}. Shamgar Gurevich (Madison) Categorical Weil Representation May 16, / 12
10 (I) Canonical Vector Space - CONSTRUCTION Heisenberg group H = V k, (v, z) (v, z ) = (v + v, z + z ω(v, v )). Additive character 1 = ψ : Z (H) = k C. Oriented Lagrangians OLag = {L = (L, o L ); L Lag(V ), o L L 0}. Irreducible rep n of H with central character ψ H L = {f : H C; f (l z h) = ψ(z)f (h) for l L, z Z, h H }. Shamgar Gurevich (Madison) Categorical Weil Representation May 16, / 12
11 Canonical Vector Space - CONSTRUCTION Vector bundle Sp H, H L = H L. OLag Shamgar Gurevich (Madison) Categorical Weil Representation May 16, / 12
12 Canonical Vector Space - CONSTRUCTION Vector bundle Sp H, H L = H L. OLag Theorem (Strong S-vN, G Hadani 04 ) We have a natural Sp-equivariant trivialization: {T M,L : H L H M } with T N,M T M,L = T N,L, for every N, M, L. Shamgar Gurevich (Madison) Categorical Weil Representation May 16, / 12
13 Canonical Vector Space - CONSTRUCTION Vector bundle Sp H, H L = H L. OLag Theorem (Strong S-vN, G Hadani 04 ) We have a natural Sp-equivariant trivialization: {T M,L : H L H M } with T N,M T M,L = T N,L, for every N, M, L. Canonical vector space H(V ) = { (f L H L, L OLag) with T M,L (f L ) = f M }. Shamgar Gurevich (Madison) Categorical Weil Representation May 16, / 12
14 Canonical Vector Space - KERNELS Kernels { C(M\H/L, ψ) HomH (H L, H M ), K M,L T M,L. Shamgar Gurevich (Madison) Categorical Weil Representation May 16, / 12
15 Canonical Vector Space - KERNELS Kernels Function of kernels { C(M\H/L, ψ) HomH (H L, H M ), K M,L T M,L. { K C(OLag 2 H), K K = K. Shamgar Gurevich (Madison) Categorical Weil Representation May 16, / 12
16 Canonical Vector Space - KERNELS Kernels Function of kernels Canonical vector space { C(M\H/L, ψ) HomH (H L, H M ), K M,L T M,L. { K C(OLag 2 H), K K = K. H(V ) = { f C(OLag H) with K f = f }. Shamgar Gurevich (Madison) Categorical Weil Representation May 16, / 12
17 (II) Geometric Kernels - DEFINITION Theorem (Geometrization, G-Hadani 06)) There exists a geometrically irreducible, perverse, l-adic Weil sheaf }{{} K on OLag 2 H with sheaf of kernels 1 Convolution. Canonical isomorphism θ : K K K. 2 Function. We have }{{} f K sheaf-to-function = K. Shamgar Gurevich (Madison) Categorical Weil Representation May 16, / 12
18 Geometric Kernels - SIGN PROBLEM Consider the commutative diagram with scalar morphism C = c Id (K K) K θ id K K θ α K (K K) id θ K K θ K C K Shamgar Gurevich (Madison) Categorical Weil Representation May 16, / 12
19 Geometric Kernels - SIGN PROBLEM Consider the commutative diagram with scalar morphism C = c Id (K K) K θ id K K θ α K (K K) id θ K K θ K C K Problem (The sign problem, Bernstein Deligne) Compute the scalar c =?. Shamgar Gurevich (Madison) Categorical Weil Representation May 16, / 12
20 Sign Problem - SOLUTION Theorem (G Hadani 11, with Gabber) We have c = 1. Shamgar Gurevich (Madison) Categorical Weil Representation May 16, / 12
21 Sign Problem - SOLUTION Theorem (G Hadani 11, with Gabber) We have c = 1. Proof. ((K K) K) K α id α (K K) (K K) α (K (K K)) K K (K (K K)) α K ((K K) K) Id id α K (K (K K)) Shamgar Gurevich (Madison) Categorical Weil Representation May 16, / 12
22 Sign Problem - SOLUTION Theorem (G Hadani 11, with Gabber) We have c = 1. Proof. ((K K) K) K α id α (K K) (K K) α (K (K K)) K K (K (K K)) α K ((K K) K) Id id α K (K (K K)) By successive application of θ, each term is identified with K, and by naturality of α, the arrows become C. Shamgar Gurevich (Madison) Categorical Weil Representation May 16, / 12
23 Sign Problem - SOLUTION Theorem (G Hadani 11, with Gabber) We have c = 1. Proof. ((K K) K) K α id α (K K) (K K) α (K (K K)) K K (K (K K)) α K ((K K) K) Id id α K (K (K K)) By successive application of θ, each term is identified with K, and by naturality of α, the arrows become C. Hence C 3 = C 2, so C = 1. Shamgar Gurevich (Madison) Categorical Weil Representation May 16, / 12
24 (III) Canonical Category - DEFINITION Definition We define C(V) = (F, η), F D b (OLag H), η : K F F, such that η is compatible with α and θ, i.e., the following diagram is commutative α (K K) F K (K F) θ id K F η F id id η K F η We call C(V) the canonical category associated with V Symp. F Shamgar Gurevich (Madison) Categorical Weil Representation May 16, / 12
25 THANK YOU Ronny Shamgar Gurevich (Madison) Categorical Weil Representation May 16, / 12
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