IMRN International Mathematics Research Notices No

38 pages

English

Découvre YouScribe en t'inscrivant gratuitement

Je m'inscris

IMRN International Mathematics Research Notices No

profil-zyak-2012 - Sergey Lysenko

Découvre YouScribe en t'inscrivant gratuitement

Je m'inscris

Obtenez un accès à la bibliothèque pour le consulter en ligne
En savoir plus

38 pages

English

Obtenez un accès à la bibliothèque pour le consulter en ligne
En savoir plus

A propos
Informations
Extrait

Description

Niveau: Supérieur, Doctorat, Bac+8
IMRN International Mathematics Research Notices 2005,No. 43 Geometric Bessel Models for GSp4 andMultiplicity One Sergey Lysenko 1 Introduction 1.1 Classical Bessel models In this paper, which is a sequel to [6], we study Bessel models of representations of GSp4 in the framework of the geometric Langlands program. These models introduced by Novodvorsky and Piatetski-Shapiro, satisfy the following multiplicity one property (see [8]). Set k = Fq and O = k[[t]] ? F = k((t)). Let ˜F be an etale F-algebra with dimF(˜F) = 2 such that k is algebraically closed in ˜F. Write ˜O for the integral closure of O in ˜F. We have two cases: (i) ˜F ?˜ k((t1/2)) (nonsplit case), (ii) ˜F ?˜ F ? F (split case). Write L for ˜O viewed as O-module, it is equipped with a quadratic form s : Sym2 L ? O given by the determinant. Write ? O for the completed module of relative di?erentials of O over k. Set M = L?(L????1 O ). This O-module is equippedwith a symplectic form ?2M? L ? L? ? ??1 O ? ??1 O .

res? pr ???????

fg ?

k? ¯q?

let z

˜x ?

projection pr

group schemes

scheme over

write ?

Informations

Publié par	profil-zyak-2012
Nombre de lectures	23
Langue	English

Extrait

IMRNInternational Mathematics Research Notices 2005,No. 43

Geometric Bessel Models forGSp4and Multiplicity One

Introduction

1.1 Classical Bessel models

Sergey Lysenko

In this paper,which is a sequel to[6],we study Bessel models of representations of GSp4the framework of the geometric Langlands program. These models introducedin by Novodvorsky and Piatetski-Shapiro,satisfy the following multiplicity one property

(see[8]).   Setk=FqandO=k[[t]]⊂F=k((t)). LetFanbeta´eelF-algebra with dimF(F)=2    such thatkis algebraically closed inF. WriteOfor the integral closure ofOinF. We have two cases:  (i)F→k((t1/2))(nonsplit case),  (ii)F→F⊕F(split case). :Sym2→O WriteLforOviewed asO-module,it is equipped with a quadratic forms L given by the determinant. WriteΩOfor the completed module of relative diﬀerentials of Ooverk. SetM=L⊕(L∗⊗ΩO−1). ThisO-module is equipped with a symplectic form∧2M→ L⊗L∗⊗ΩO−1→ΩO−1. SetG=GSp(M),this is a group scheme over SpecO. WriteP⊂G for the Siegel parabolic subgroup preserving the Lagrangian submoduleL. Its unipotent radicalUhas a distinguished character

ev:U−→ΩO⊗Sym2Ls→ΩO −

Received 5 January 2005. Revision received 19 May 2005. Communicated by Edward Frenkel.

(1.1)

2658

Sergey Lysenko

(here we viewΩOas a commutative group scheme over SpecO). Set

R=p∈P|evpup−1=ev(u)foru∈U.

(1.2)

 View GL(L)as a group scheme over SpecOandO∗as its closed subgroup. Writeαfor the   compositionO∗→GL(L)d→etO∗. Fix a sectionO∗→Rgiven byg→(g, α(g)(g∗)−1). Then R=O∗U⊂Ris a closed subgroup,and the mapRξ→ΩO×O∗sendingtuto(ev(u), t)is a homomorphism of group schemes over SpecO.   ¯ Letbe a prime invertible ink. Fix a characterχ:F∗/O∗→Q∗and a nontrivial ¯ additive characterψ:k→Q∗. Writeτfor the composition

¯ R(F)−ξ→ΩF×F∗−Re−s−×−p−r→k×F∗/O∗−ψ−×−χ→Q∗.

The Bessel module is the vector space

BMτ=f:G(F)/G(O)−→Q¯|f(rg)=τ(r)f(g)forr∈R(F), fis of compact support moduloR(F).

¯ Letχc:F∗/O∗→Q∗denote the restriction ofχ. The Hecke algebra Hχc=h:G(O)\G(F)/G(O)−→Q¯|h(zg)=χc(z)h(g)forz∈F∗, his of compact support moduloF∗

(1.3)

(1.4)

(1.5)

acts on BMτby convolutions. Then BMτisa free module of rank oneover Hχc. In this paper we prove a geometric version of this result. Recall that the aﬃne Grassmannian GrG=G(F)/G(O)can be viewed as an ind-scheme overk. According to “fonctions-faisceaux” philosophy,the space BMτshould have a geometric counterpart. A natural candidate for that would be the category of-adic perverse sheaves on GrGthat change under the action ofR(F)byτ. However,theR(F)-orbits on GrGare inﬁnite-dimensional,and this naive deﬁnition does not make sense. The same diﬃculty appears when one tries to deﬁne Whittaker categories for any reductive group. In[3]Frenkel,Gaitsgory,and Vilonen have overcome this by replacing the corresponding local statement by its globalization,which admits a geometric counterpart leading to a deﬁnition of Whittaker categories with expected properties. We follow the strategy of[3]replacing the above local statement by a global one,which we further geometrize.

1.2 Geometrization

Geometric Bessel Models forGSp4and Multiplicity One

2659

 Fix a smooth projective absolutely irreducible curveXoverk. Letπ:X→Xbe a two- sheeted covering ramiﬁed at some eﬀective divisorDπofX(we assumeXsmooth overk). The vector bundleL=π∗OXis equipped with a quadratic forms:Sym2L→OX. WriteΩfor the canonical line bundle onX. SetM=L⊕(L∗⊗Ω−1),it is equipped with a symplectic form

−1 ∧2M−→L⊗L∗⊗Ω−1−→Ω .

(1.6)

LetGbe the group scheme(overX)of automorphisms ofMpreserving this symplec-tic form up to a multiple. LetP⊂Gdenote the Siegel parabolic subgroup preservingL, U⊂Pits unipotent radical. ThenUis equipped with a homomorphism of group schemes overX

ev:U−→Ω⊗Sym2L−s→Ω.

(1.7)

LetTbe the functor sending aX-schemeSto the group H0(X×XS,O∗). ThenTis a group scheme overX,a subgroup of GL(L). Writeαfor the compositionT→GL(L)d→etGm. Set R=p∈P|evpup−1=ev(u)∀u∈U.(1.8)   Fix a sectionT→Rgiven byg→(g, α(g)(g∗)−1). ThenR=TU⊂Ris a closed subgroup, and the mapRξ→Ω×Tsendingtuto(ev(u), t)is a homomorphism of group schemes overX.

LetF=k(X),letAbe the adele ring ofF,andO⊂Athe entire adeles. WriteFx for the completion ofFatx∈XandOx⊂Fxfor its ring of integers. Fix a nonramiﬁed ¯ characterχ:T(F)\T(A)/T(O)→Q∗. Letτbe the composition R(A)−ξ→Ω(A)×T(A)−r−×−χ→Q¯∗,(1.9) ¯ wherer:Ω(A)→Q∗is given by r(ωx)=ψxtrk(x)/kResωx.(1.10) ∈X Fixx∈X(k). LetYdenote the restricted productG(Fx)/G(Ox)×y=xR(Fy)/R(Oy). LetY(k)be the quotient ofYby the diagonal action ofR(F). Set ¯ BMX,τ=f:Y−→Q|f(rg)=poτr(tr)fm(ogd)forur∈R(A),(1.11) fis of compact sup loR(A).

Univers
Ebooks
Livres audio
Presse
Podcasts
BD
Documents

IMRN International Mathematics Research Notices No

YouScribe

Le catalogue

Le service

Les conditions