Geometric Weil representation in characteristic two

Alain Genestier

Description

ar X iv :0 90 6. 06 98 v1 [ ma th. RT ] 3 J un 20 09 Geometric Weil representation in characteristic two Alain Genestier and Sergey Lysenko Abstract Let k be an algebraically closed field of characteristic two. Let R be the ring of Witt vectors of length two over k. We construct a group stack G over k, the metaplectic extension of the Greenberg realization of Sp2n(R). We also construct a geometric analog of the Weil representation of G, this is a triangulated category on which G acts by functors. This triangulated category and the action are geometric in a suitable sense. 1. Introduction 1.1 Apparently, a version of the Weil representation in characteristic two first appeared in 1958 paper by D. A. Suprunenko ([22], Theorem 11) (before the celebrated 1964 paper by A. Weil [24]). This representation and its character were also studied in [16, 15]. Being inspired mostly by [14] and [17], in this paper we propose a geometric analog of this representation. Let k be a finite field of characteristic two. Let R be the ring of Witt vectors of length 2 over k. Given a free R-module V˜ of rank 2n with symplectic form ?˜ : V˜ ? V˜ ? R, set V = V˜ ?R k.

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• heisenberg group


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• forms

Sujets

• Unipotent
• Heisenberg group

Informations

• Publié par: pefav
• Nombre de lectures: 5
• Langue: English

Extrait

Geometric

Weil representation in characteristic two

Alain Genestier and Sergey Lysenko

AbstractLetk Letbe an algebraically closed field of characteristic two.Rbe the ring ˆ of Witt vectors of length two overk construct a group stack. WeGoverk, the metaplectic extensionoftheGreenberg ˆ realizationofSp 2n (Rnocourtsgatcemoe.W)lseatricanalogof ˆ the Weil representation ofG, this is a triangulated category on whichGacts by functors. This triangulated category and the action are geometric in a suitable sense.

1. Introduction

1.1 Apparently, a version of the Weil representation in characteristic two first appeared in 1958 paper by D. A. Suprunenko ([22], Theorem 11) (before the celebrated 1964 paper by A. Weil [24]). This representation and its character were also studied in [16, 15]. Being inspired mostly by [14] and [17], in this paper we propose a geometric analog of this representation. Letk Letbe a finite field of characteristic two.RWitt vectors of length 2 overbe the ring of ˜ ˜ ˜ ˜ k. Given a freeR-moduleVof rank 2nwith symplectic form ˜ω:V×V→R, setV=V⊗ R k. ˜ ˜  ˜  ˜ ˜  ˜ Pick a bilinear formβ:V×V→Rsuch thatβ(˜x, y˜)−β(y˜, x˜) =ω˜(x˜, y˜) for allx˜, y˜∈V. Let ˜ ˜ β:V×V→2R⊂Rbe the map (x, y)7→2β(˜x, y) for anyx, y∈Voverx, y∈V. It gives rise ˜ ˜ ˜ to the Heisenberg groupH(V) =V×Rwith operation

(v 1 , z 1 )(v 2 , z 2 ) = (v 1 +v 2 , z 1 +z 2 +β(v 1 , v 2 )) The reason for usingRinstead of 2Rin the definition ofH(V) is that in this way it aquires a ˜ larger group of automorphisms acting trivially on the center. The groupSp(V) maps naturally to this group. ¯ Fix a primeℓ6= 2 and a faithful characterψ:Z4Z→Q ∗ ℓ . A version of the Stone-von ˜ Neumann theorem holds in this setting giving rise to the metaplectic extension ofSp(V) and its Weil representationH ψ  According to [14], it can be seen as a group(cf. Section 2 for details). ˜ Mp(V) that fits into an exact sequence ¯ ˜ ˜ 1→ 2 (Q ℓ )→Mp(V)→Sp(V)→1 (1) In the geometric setting, assumekto be an algebraically closed field of characteristic two. ˜ ˜ We propose geometric analogs of Mp(V) andH ψ . LetVbe a freeR-module of rank 2nwith a ˜ symplectic form. WriteGfor the Greenberg realization of theR-schemeSp(V). ViewH(V) as a group scheme overk, an extension ofVby the Greenberg realization ofR. Though we mostly follow the strategy of [17], there are new difficulties and phenomena in characteristic two. To the difference with the case of other characteristics, the metaplectic

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ˆ extension (1) is nontrivial. The geometric analog of (1) is an algebraic group stackGoverk that fits into an exact sequence ˆ 1→B(Z4Z)→G→G→1

(2)

of group stacks overkour Remark 8 ii) it follows that there is a group stack from . Actually, ˆ ˆ G b overkincluded into an exact sequence 1→B(Z2Z)→G b →G→1 such that (2) is its ˆ push-forward via the natural mapB(Z2Z)→B(Z4Z properly,). MoreG b is the geometric ˜ ˆ analog of Mp(V), butG b will not appear in this paper. ˆ We don’t know ifGadmits a presentation as the stack quotientG 1 G 0 for a morphism G 0 →G 1 of algebraic groups overk, whereG 0 is abelian and maps to the center ofG 1 (we ˆ would rather expect thatGcorresponds to a nontrivial crossed module). We have not found a relation with the K-theory (or the universal central extension ofSp 2n ˆ byK 2 constructed by Brylinski-Deligne [8]). our construction of Instead,Ggoes as follows. ˜ ˜ LetL(V) be the Greenberg realization of theR-scheme of free lagrangian submodules inV. First, we define certainZ4Z-gerb ˆ ˜ ˜ L(V)→ L(V) via the geometric Maslov index (cf. Section 6). It turns out that the corresponding class in ˜ ˆ H 2 (L(V),Z4Z) is invariant underG, but the gerb itself is notG-equivariant. ThenGis defined ˆ ˜ ˆ ˜ as the stack of pairs (g, σ), whereg∈Gandσ:g ∗ L(V)→fL(V) is an isomorphism ofZ4Z-gerbs ˜ overL(V). We generalize the theory of canonical interwining operators from [17] to the case of charac- teristic two (cf. Section 7). This allows us to come up with a construction of the Weil category ˜ ˜ W(V), which is a geometric analog ofH ψ . HereW(V) is a category of certain perverse sheaves ˆ ˜ ˆ ˜ onL(V)×H(V group stack). TheGacts onW(V This action is geometric in the) by functors. ˆ ˆ ˜ sense that it comes from the natural action ofGonL(V)×H(V). Similarly to the case of other characteristics, we also constructthe finite-dimensional theta- sheafS V ˜ ψ , which is a geometric analog of some matrix coefficient of the representationH ψ . It ˜ serves as the key ingredient for the construction ofW(V). Acknowledgements.We are very grateful to P. Deligne for his comments and suggestions about the first version of this paper. 1.2NotationLetAbe a commutative ring andVa freeA-module of rankd. As in ([12], Section 5.1), denote by Sym !2 (V)⊂V⊗Vthe submodule ofS 2  It-invariant vectors. is the submodule spanned by the vectors of the formv⊗v,v∈V. Let∧ 2 (V)⊂V⊗Vbe the submodule spanned by the vectors of the formv⊗u−u⊗v Sym. Let ∗2 (V) be the quotient of V⊗Vby∧ 2 (V). For anynlet ∧ n (V) = 1≤i≤ ∩ n1 V ⊗i−1 ⊗(∧ 2 V)⊗V ⊗d−1−i − For anynwrite Sym !n (V)⊂V ⊗n for the submodule ofS n -invariant tensors and Sym ∗n (V) for theA-module ofS n -coinvariants ofV ⊗n . We have canonically Sym !n (V ∗ )→f(Sym ∗n (V)) ∗ and∧ n (V ∗ )f→(∧ n V) ∗

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The space Sym !2 (V ∗ can be seen as the space of symmetric bilinear forms on) V is the. This space ofA-linear mapsφ:V→V ∗ such thatφ ∗ =φ. LetVbe a freeA-module of rank 2n. Say thatVis a symplectic if it is equipped with a non-degenerate bilinear formω:V×V→Asuch that in a suitable base (e i , e −i ,1≤i≤n) we haveω(e i , e −j ) =δ ij ,ω(e i , e j ) =ω(e −i , e −j ) = 0 fori, j∈ {1,    , n}andω(x, y) =−ω(y, x). We call such basea symplectic base. IfVis a symplecticA-module then we trivializeω ∧ : detVf→A bye7→1, wheree=e 1 ∧  ∧e n ∧e −1 ∧  ∧e −n does not depend on a symplectic base (e i , e −i ). Forω n ∈ ∧ 2n M ∗ we haven!e=±ω n . The pairing between a freeA-module and its dual is usually denoted byh, i. IfAis of characteristicp, writeV (p) =V⊗ A A, whereA→A,a7→a p is the Frobenius map. 1.3Generalities on quadratic formsLetkof characteristic two, which is eitherbe a field finite or algebraically closed. LetRbe the ring of Witt vectors of length two overk. LetLbe a finite-dimensionalk Let-vector space.B a (L ∗ )⊂(L⊗L) ∗ be the subspace of bilinear formsφonLsftingyisaφ(x, x) = 0 for allx∈L call them. Wealternating bilinear forms. WriteQ(L ∗ ) for thek-space of quadratic forms onL definition, it is included into an. By exact sequence 0→B a (L ∗ )→(L⊗L) ∗ →Q(L ∗ )→0, where the second map sends a bilinear formφto the quadratic formx7→φ(x, x). WriteQ a (L ∗ ) for thek-space of additive quadratic forms onL map. TheL ∗ →Q a (L ∗ ) sendingy ∗ to the quadratic formy7→ hy, y ∗ i 2 ,y∈Lyields an isomorphism (L ∗ ) (2) f→Q a (L ∗ ). One has an exact sequence 0→B a (L ∗ )→Sym !2 (L ∗ )→Q a (L ∗ )→0 ofk-vector spaces. One also has an exact sequence

0→Q a (L ∗ )→Q(L ∗ )→B a (L ∗ )→0, where the second map sendsqto the the bilinear form (x, y)7→q(x+y)−q(x)−q(y). ˜ ˜ Assume given a freeR-moduleLwith an isomorphismL⊗ R k→fLofk-vector spaces. Note thatB a (L ∗ )⊂Sym !2 ( ˜ L ∗ ) can be seen as theR-submodule consisting ofϕgniysasftiϕ(x˜, x˜) = 0 ˜ ! (L ∗ ) be the quotient of Sym !2 ( ˜ L ∗ ) byB a (L ∗ ), this is anR-module included for all ˜x∈L. LetQ into an exact sequence 0→Q a (L ∗ )→Q ! (L ∗ )→Sym !2 (L ∗ )→0 ofR action of-modules. The ˜ GL(L) onQ ! (L ∗ ) factors through an action of GL(L). Equivalently, one may defineQ ! (L ∗ ) as theR-module of mapsq:L→Rsuch that

•the mapb q :L×L→Rgiven byb q (x 1 , x 2 ) =q(x 1 +x 2 )−q(x 1 )−q(x 2 ) is bi-additive, andb q (ax 1 , x 2 ) = ˜ab q (x 1 , x 2 ) for anya˜∈Rovera∈k; •q(ax) =a˜ 2 q(x) forx∈Land ˜a∈Rovera∈k.

TheseR-valued ‘quadratic forms’ onLhave been considered, for example, in [14, 25, 18].

2. Classical Weil representation and motivations

2.1. In this section we remind the construction of the Weil representation in characteristic two following essentially [14]. This is our subject to geometrize.

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2.2. Letkbe a finite field of characteristic two withqelements,Rbe the ring of Witt vectors of ˜ ˜ ˜ length two overk. LetVbe a freeR-module of rank 2nwith symplectic formω˜ :V⊗V→R. ˜ ˜ SetV=V⊗ R k. Letω:V×V→2Rbe given byω(x, y) = 2ω˜(˜x, y˜) for anyx˜, y˜∈Vover x, y∈V. ˜ ˜  ˜ Pick a bilinear formβ:V×V→Rsuch that

˜ ˜ β(˜x, y˜)−β(y˜,˜x) = ˜ω(x˜, y˜) (3) ˜ ˜  ˜ for all ˜x, y˜∈V. Letβ:V×V→Rbe the map (x, y)7→2β(x˜, y˜) for anyx˜, y˜∈Voverx, y. It gives rise to the Heisenberg groupH(V) =V×Rwith operation

(v 1 , z 1 )(v 2 , z 2 ) = (v 1 +v 2 , z 1 +z 2 +β(v 1 , v 2 )), v i ∈V, z i ∈R(4) Its center isZ(H(V)) ={(0, z)∈H(V)|z∈R}. Gurevich and Hadani consider the group of all automorphisms ofH(V) acting trivially on the centerZ(H(V)). For the purposes of geometrisation, we modify their definition slightly as follows. Let ASp(V) be the set of pairs (g, α), whereg∈Sp(V) andα:V→Rsatisfies

•α(v 1 +v 2 )−α(v 1 )−α(v 2 ) =β(g(v 1 ), g(v 2 ))−β(v 1 , v 2 ) for allv i ∈V; •α(av) =a˜ 2 α(v) for anyv∈Vand ˜a∈Rovera∈k.

An element (g, α)∈ASp(V) yields an automorphism ofH(V) given by (v, z)7→(gv, z+α(v)). In this way ASp(V) maps injectively into the group of automorphisms ofH(V) acting trivially onZ(H(V composition in A)). TheSp(V) is given by (g, α g )(h, α h ) = (gh, h −1 (α g ) +α h ) withh −1 (α g )(v) =α g (hv) for allv∈V will refer to A. WeSp(V) asthe affine symplectic group. For ak-vector spaceLwrite F Q a (L ∗ ) ={α:L→R|α(x 1 +x 2 ) =α(x 1 )+α(x 2 ) andα(ax) = ˜a 2 α(x),˜a∈Rovera∈k, x∈L} An element ofF Q a (L ∗ writes in Witt coordinates as (0) , α 1 ), whereα 1 :L→kis additive and α 1 (ax) =a 4 α 1 (x) for alla∈k, x∈L. So,F Q a (L ∗ )f→(L ∗ ) (4) . The group ASp(V) fits into an exact sequence 1→F Q a (V ∗ )→ASp(V)→Sp(V)→1 ˜ Though it is not reflected in the notation, ASp(V) depends not only onω˜ but also onβ. ˜ LetG=Sp(V). We have a surjective homomorphismξ:G→ASp(V) sending ˜gto (g, α g˜ ), whereg∈Sp(V) is the image ofg˜, andα g˜ :V→Ris given by ˜ ˜ α g˜ (v) =β(˜g˜ ˜gv˜)−β(v˜,˜v) v,

˜ for any ˜v∈Voverv∈V.

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