Moduli of metaplectic bundles on curves and theta-sheaves

4^esérie, t. 39, 2006, p. 415 à 466.

MODULI OF METAPLECTIC BUNDLES ON CURVES AND THETA-SHEAVES

B

S

LYSENKO

ABSTRACT. – We give a geometric interpretation of the Weil representation of the metaplectic group, placing it in the framework of the geometric Langlands program.

For a smooth projective curveXwe introduce an algebraic stackBunGof metaplectic bundles onX. It also has a local versionGrG, which is a gerbe over the affine Grassmanian ofG. We define a categorical version of the (nonramified) Hecke algebra of the metaplectic group. This is a categorySph(GrG)of certain perverse sheaves onGrG, which act onBunGby Hecke operators. A version of the Satake equivalence is proved describingSph(GrG) as a tensor category. Further, we construct a perverse sheaf onBunG

corresponding to the Weil representation and show that it is a Hecke eigen-sheaf with respect toSph(GrG).

©2006 Elsevier Masson SAS

RÉSUMÉ. – On donne une interprétation géométrique de la représentation de Weil du groupe métaplectique, qui s’inscrit dans le cadre du programme de Langlands géométrique.

Pour une courbeX lisse projective on introduit un champ algébriqueBunGdes fibrés métaplectiques surX. Il admet aussi une version locale GrG, qui est une gerbe sur la grassmanienne affine deG. On définit une version catégorique de l’algèbre de Hecke (non ramifiée) du groupe métaplectique. C’est une catégorieSph(GrG)de certains faisceaux pervers surGrG, qui agissent surBunGpar les opérateurs de Hecke. On démontre une version de l’équivalence de Satake qui décrit la catégorie tensorielleSph(GrG).

Ensuite, on construit un faisceau pervers surBunGqui correspond à la représentation de Weil et on établit sa propriété de Hecke par rapport àSph(GrG).

©2006 Elsevier Masson SAS

1. Introduction 1.1. Historically θ-series (such as, in one variable,

qⁿ²) have been one of the major methods of constructing automorphic forms. A representation-theoretic approach to the theory of θ-series, as discovered by A. Weil [23] and extended by R. Howe [15], is based on the oscillator representation of the metaplectic group (cf. [22] for a recent survey). In this paper we propose a geometric interpretation of this representation (in the non-ramified case) placing it in the framework of the geometric Langlands program.

Let k=Fq be a finite field withq odd. Set K=k((t))and O=k[[t]]. Let Ω denote the completed module of relative differentials of O over k. Let M be a freeO-module of rank 2ngiven with a non-degenerate symplectic form2

M →Ω. It is known that the continuous cohomology groupH²(Sp(M)(K),{±1})is isomorphic toZ/2Z[19, 10.4]. AsSp(M)(K)is

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE

a perfect group, the corresponding metaplectic extension

1→ {±1}−→ⁱ Sp(M)(K)→Sp(M)(K)→1 (1)

is unique up to unique isomorphism¹. It can be constructed in two essentially different ways.

Recall the classical construction of A. Weil [23]. The Heisenberg group isH(M) =M⊕Ω with operation

(m1, ω1)(m2, ω2) =

m1+m2, ω1+ω2+1

2m1, m2

.

Fix a prime that does not divide q. Let ψ:k→Q^∗ be a nontrivial additive character. Let χ: Ω(K)→Qbe given byχ(ω) =ψ(Resω). By the Stone and Von Neumann theorem [21], there is a unique (up to isomorphism) smooth irreducible representation(ρ,Sψ)ofH(M)(K) overQ with central characterχ. The groupSp(M)acts onH(M)by group automorphisms (m, ω)−→^g (gm, ω)This gives rise to the group

Sp(M)(K) = g, M[g]

|g∈Sp(M)(K), M[g]∈AutSψ

ρ(gm, ω)◦M[g] =M[g]◦ρ(m, ω) for(m, ω)∈H(M)(K)

.

The groupSp(M )(K) is an extension of Sp(M)(K)by Q^∗. Its commutator subgroup is an extension ofSp(M)(K)by{±1}→Q^∗, uniquely isomorphic to (1).

Another way is via Kac–Moody groups. Namely, viewSp(M)(K)as an ind-scheme overk.

Let

1→Gm→Sp(M)(K)→Sp(M)(K)→1 (2)

denote the canonical extension, hereSp(M)(K)is an ind-scheme overk(cf. [12]). Passing to k-points we get an extension of abstract groups1→k^∗→Sp(M)(K)→Sp(M)(K)→1. Then (1) is the push-forward of this extension underk^∗→k^∗/(k^∗)².

The second construction underlies one of our main results, the Tannakian description of the Langlands dual to the metaplectic group. Namely, the canonical splitting of (2) overSp(M)(O) yields a splitting of (1) overSp(M)(O). Consider the Hecke algebra

H= f:Sp(M)(O)\Sp(M)(K)/Sp(M)(O)→Q|f

i(−1)g

=−f(g), g∈Sp(M)(K);

fis of compact support .

The product is convolution, defined using the Haar measure onSp(M)(K)for which the inverse image ofSp(M)(O)has volume1.

SetG=Sp(M). LetGˇdenoteSp_2nviewed as an algebraic group overQ. LetRep( ˇG)denote the category of finite-dimensional representations ofG. Writeˇ K(Rep( ˇG))for the Grothendieck ring ofRep( ˇG)overQ. There is a canonical isomorphism ofQ-algebras

H→K

Rep( ˇG) .

Actually, a categorical version of this isomorphism is proved. Consider the affine Grassmanian GrG=G(K)/G(O), viewed as an ind-scheme over k. Let W denote the nontrivial -adic

1The notation_Sp(M)(K)is ambiguous, these are notK-points of an algebraic group.

e ◦

local system of rank one onGm corresponding to the covering Gm→Gm,x→x². Denote bySph(GrG)the category ofG(O)-equivariant perverse sheaves onG(K)/G(O), which are also(Gm, W)-equivariant. HereGrG denotes the stack quotient ofG(K)/G(O)byGm with respect to the actiong−→^x x²g,x∈Gm, g∈G(K). Actually,Sph(GrG)is a full subcategory of the category of perverse sheaves onGr_G.

Replacing for simplicity k by an algebraically closed field, we equip Sph(GrG) with the structure of a rigid tensor category. We establish a canonical equivalence of tensor categories

Sph(GrG)→Rep( ˇG).

1.2. In the global setting letXbe a smooth projective curve overk. LetGdenote the sheaf of automorphisms ofO_Xⁿ⊕Ωⁿ(nowΩis the canonical line bundle onX) preserving the symplectic form₂

(Oⁿ_X⊕Ωⁿ)→Ω. The stackBun_GofG-bundles (=G-torsors) onX classifies vector bundlesM of rank 2non X, given with a non-degenerate symplectic form₂

M →Ω. We introduce an algebraic stackBun_Gof metaplectic bundles onX. The stackGr_Gis a local version ofBunG. The categorySph(GrG)acts onD(BunG)by Hecke operators.

We construct a perverse sheafAutonBun_G, a geometric analog of the Weil representation.

We calculate the fibres ofAut and its constant terms for maximal parabolic subgroups ofG.

Finally, we argue thatAutis a Hecke eigensheaf onBunGwith eigenvalue St = RΓ

P²ⁿ⁻¹,Q

⊗Q[1]

1 2

_⊗2n−1

viewed as a constant complex onX. Note thatStis equipped with an action ofSL₂of Arthur, the corresponding representation ofSL₂is irreducible of dimension2nand admits a unique, up to a multiple, symplectic form. One may imagine thatAutcorresponds to a group homomorphism π1(X)×SL2→Gˇtrivial onπ1(X). This agrees with Arthur’s conjectures.

2. Weil representation and motivations

2.1. LetXbe a smooth projective absolutely irreducible curve overk=Fq,F=Fq(X),Abe the adeles rings ofF,O ⊂Abe the entire adeles. Assume thatqis odd. Fix a primethat does not divideq. LetΩdenote the canonical line bundle onX.

LetM be a2n-dimensional vector space overF with symplectic form2

M→ΩF, where ΩF is the generic fibre ofΩ. The Heisenberg groupH(M) =M⊕ΩF with operation

(m1, ω1)(m2, ω2) =

m1+m2, ω1+ω2+1

2m1, m2

is algebraic over F. Fix a nontrivial additive character ψ:Fq → Q^∗. Then H(M)(A) = M(A)⊕Ω(A)admits a canonical central characterχ: Ω(A)/Ω(F)→Q^∗ given by

χ(ω) =ψ

x∈X

tr_k(x)/kResωx

.

The Stone and Von Neumann theorem [21] says that there is a unique (up to isomorphism) smooth irreducible representation(ρ,Sψ)ofH(M)(A)overQ with central characterχ. The

groupSp(M)acts on H(M) by group automorphisms(m, ω)−→^g (gm, ω). This defines the global metaplectic group

Sp(M )(A) = g, M[g]

|g∈Sp(M)(A), M[g]∈AutSψ,

ρ(gm, ω)◦M[g] =M[g]◦ρ(m, ω)for(m, ω)∈H(M)(A) included into an exact sequence

1→Q^∗→Sp(M )(A)→Sp(M)(A)→1.

(3)

The representation ofSp(M)(A)onSψis called the Weil (or oscillator) representation [23].

For a subgroupK⊂Sp(M)(A)writeK for the preimage of K inSp(M )(A). Sinceχ is trivial on ΩF, one may talk aboutH(M)-invariant functionals onSψ, they are called theta- functionals. The space of theta-functionals is1-dimensional and preserved bySp(M)(F), so the action ofSp(M)(F)on this space defines a splitting of (3) overSp(M)(F).

View

Funct

Sp(M)(F)\Sp(M)(A)

= f:Sp(M)(F)\Sp(M)(A)→Q

as a representation ofSp(M )(A)by right translations. A theta-functionalΘ :Sψ→Qdefines a morphism ofSp(M)(A)-modules

Sψ→Funct

Sp(M)(F)\Sp(M )(A) (4)

sendingφtoθ_φgiven byθ_φ(g) = Θ(gφ)forg∈Sp(M)(A).

Now assume that M is actually a rank 2n vector bundle on X with symplectic form ₂

M →Ω. Then we get the subgroups Sp(M)(O)⊂Sp(M)(A) and M(O)⊕Ω(O)⊂ H(M)(A). Moreover, the space of M(O)⊕Ω(O)-invariants in Sψ is 1-dimensional and preserved bySp(M )(O). The action ofSp(M )(O)on this space yields a splitting of (3) over Sp(M)(O). Ifφ0∈ Sψis a nonzeroM(O)⊕Ω(O)-invariant vector then its image under (4) is the classical theta-function

f0:Sp(M)(F)\Sp(M )(A)/Sp(M)(O)→Q

that we are going to geometrize.

LetGdenote the sheaf of automorphisms ofM preserving the form ₂

M →Ω. This is a sheaf of groups (in flat topology) onXlocally in Zariski topology isomorphic toSp_2n.

2.2. AssumeM=V ⊕(V^∗⊗Ω)is a direct sum of Lagrangian subbundles, the form being given by the canonical pairing·,·betweenV andV^∗. Let

χ_V:V(A)⊕Ω(A)→Q^∗

denote the characterχV(v, ω) =χ(ω).

We have the subgroupV(A)⊂H(M)(A). The space ofV(A)-invariant functionals onSψ

is1-dimensional. A choice of such functional identifies Sψ with the induced representation of (V(A)⊕Ω(A), χV)toH(M)(A). The latter identifies with the Schwarz spaceS(V^∗⊗Ω(A)) of locally constant compactly supportedQ-valued functions onV^∗⊗Ω(A), the corresponding

e ◦

functional onS(V^∗⊗Ω(A))becomes the evaluation at zeroev:S(V^∗⊗Ω(A))→Q. This is the Schrödinger model ofSψ.

Writeg∈Sp(M)(A)as a matrix

g= a b

c d

, (5)

witha∈End(V)(A), b∈Hom(V^∗⊗Ω, V)(A), d∈End(V^∗)(A), c∈Hom(V, V^∗⊗Ω)(A).

Writea^∗for the transpose operator toa.

The defined up to a scalar automorphismM[g]ofS(V^∗⊗Ω(A))is described as follows.

• Fora∈GL(V)(A)we have _{a 0}

0a^∗−1

∈Sp(M)(A). Besides,

1b 0 1

∈Sp(M)(A)if and only ifb∈(V ⊗V ⊗Ω⁻¹)(A)is symmetric. Forggiven by (5) withc= 0we have

M[g]f

(v^∗) =χ 1

2a^∗v^∗, b^∗v^∗

f(a^∗v^∗), v^∗∈V^∗⊗Ω(A), (6)

• ifb:V^∗⊗Ω(A)→V(A)theng= _{0 b}

−b^∗−10

∈Sp(M)(A)and M[g]f

(v^∗) =

V(A)

χ v, v^∗

f b⁻¹v

dv, v^∗∈V^∗⊗Ω(A) (7)

for any Haar measuredvonV(A).

LetP⊂Gdenote the Siegel parabolic subgroup preservingV. The subgroupP(A)preserves evup to a multiple, so defining a splitting of (3) overP(A). This splitting coincides with the one given by (6).

Letφ0∈ S(V^∗⊗Ω(A))denote the characteristic function ofV^∗⊗Ω(O). Using (6) and (7) one shows thatφ0generates the space ofSp(M)(O)-invariants inS(V^∗⊗Ω(A)). In this model ofSψthe theta functionalΘ :S(V^∗⊗Ω(A))→Qis given by

Θ(φ) =

v^∗∈V^∗⊗Ω(F)

φ(v^∗) forφ∈ S

V^∗⊗Ω(A) .

Letf₀denote the image ofφ₀under the corresponding map (4). Let us calculate the composition P(F)\P(A)/P(O)→Sp(M)(F)\Sp(M )(A)/Sp(M)(O)−→^f0 Q

denoted by f_P. We used the fact that the splittings of (3) over P(A) and Sp(M)(O) are compatible overP(O).

Denote byBunnthek-stack of ranknvector bundles onX. The setGL(V)(A)/GL(V)(O) naturally identifies with the isomorphism classes of pairs (L, α), where L∈Bunn(k) and α:L(F)→V(F). HereL(F)is the generic fibre ofL.

Leta∈GL(V)(A)and(L, α)be the pair attached toaGL(V)(O). Then v^∗∈V^∗⊗Ω(F)|a^∗v^∗∈V^∗⊗Ω(O) _α∗

−−→Hom(L,Ω) (8)

is an isomorphism.

The group P fits into an exact sequence 1→(Sym²V)⊗Ω⁻¹ →P →GL(V)→1 of algebraic groups overX. Forg∈P(A)we get

fP(g) = Θ(gφ0) =

v^∗∈V^∗⊗Ω(F)

(gφ0)(v^∗)

=

v^∗∈V^∗⊗Ω(F)

χ 1

2a^∗v^∗, b^∗v^∗

φ₀(a^∗v^∗)

=

s∈Hom(L,Ω)

χ 1

2s, ab^∗s

in view of (8).

LetBunP be thek-stack ofP-bundles onX. ItsY-points for a schemeY is the category of (Y ×X)×XP-torsors overY ×X. ThenBun_P classifies pairs L∈Bun_n together with an exact sequence onX

0→Sym²L→?→Ω→0.

(9)

(More generally, for a semidirect product of group schemes1→U →P →M →1providing aP-torsorFP is equivalent to providing aM-torsorFM and a U_FM-torsor of isomorphisms Isom(FP,FM×M P)inducing a given one on the correspondingM-torsors.)

In view of the bijection P(F)\P(A)/P(O)→Bun_P(k), the function f_P on Bun_P(k) is described as follows. Let aP-torsorFP∈Bun_P(k)be given byL∈Bun_n(k)together with (9).

Consider the mapq^FP: Hom(L,Ω)→ksendings∈Hom(L,Ω)to the pairing of s⊗s∈Hom

Sym²L,Ω^⊗2

with the exact sequence (9). Then

fP(FP) =

s∈Hom(L,Ω)

ψ

q^FP(s) .

The functionfP: BunP(k)→Q is the trace of Frobenius of the following-adic complex SP,ψ onBunP.

Letp:X →BunP be the stack overBunP with fibreHom(L,Ω). Letq:X →A¹be the map sendings∈Hom(L,Ω)to the pairing of (9) with

s⊗s∈Hom

Sym²L,Ω^⊗2 .

The geometric analog offP is the complexSP,ψ=p!q^∗Lψ⊗Q[1](¹₂)^⊗dimX onBunP, here dimX denotes the dimension of the corresponding connected component ofX.

3. Main results 3.1. Notation

From now onkdenotes an algebraically closed field of characteristicp >2, all the schemes (or stacks) we consider are defined overk.

LetX be a smooth projective connected curve. WriteΩfor the canonical line bundle onX. Fix a prime=p. For a scheme (or stack)S writeD(S)for the bounded derived category of -adic étale sheaves onS, andP(S)⊂D(S)for the category of perverse sheaves (the middle perversity function is always taken in absolute sense overSpeck).

e ◦

Fix a nontrivial character ψ:Fp→Q^∗ and denote byLψ the corresponding Artin–Shreier sheaf onA¹. Fix a square rootQ(¹₂)of the sheafQ(1)onSpecFq. Isomorphism classes of such correspond to square roots ofqinQ. Fix an inclusion of fieldsFq→k.

If V →S andV^∗ →S are dual rank n vector bundles over a stack S, we normalize the Fourier transformFourψ: D(V)→D(V^∗)byFourψ(K) = (pV^∗)!(ξ^∗Lψ⊗p^∗_VK)[n](ⁿ₂), where pV, pV^∗ are the projections, andξ:V×SV^∗→A¹is the pairing.

AG-torsor on a schemeSis also referred to as aG-bundle onS. WriteVect^εfor the tensor category ofZ/2Z-graded vector spaces, our conventions about this category are those of [8].

WriteVect⊂Vect^εfor its even component, i.e., the tensor category of vector spaces.

3.1.1. The sheaf (in flat topology) on the category of k-schemes represented by μ2 :=

Ker(x→x²:Gm→Gm)is the constant sheaf{±1}.

For a schemeSand a line bundleAonSdenote bySthe followingμ2-gerbe overS. For an S-schemeS, the category ofS-points ofSis the category of pairs(B,B²→ A|S), whereBis a line bundle onS. Note thatS→Sis étale.

IfS→Sadmits a section given by invertibleOS-moduleB0together withB²₀→ Athen the gerbe is trivial, that is,S→B(μ₂/S)overS. In this case we get theS₂-coveringCov(S) →S, whose fibre consists of isomorphismsB→ B0 whose square is the given one B²→ A. This covering is locally trivial in étale topology, but not trivial even for S = Speck. Actually S= Cov(S).

3.1.2. If in additionA is a Z/2Z-graded line bundle onS purely of degree zero, then by definitionSclassifies aZ/2Z-graded line bundleBpurely of degree zero, given with aZ/2Z- graded isomorphismB²→ A. IfBis aZ/2Z-graded line bundle on Sof pure degree (that is, placed in one degree only over each connected component) then aZ/2Z-graded isomorphism B²→ Ayields a (uniquely defined) section ofS.

3.2. Let Bun_n be the stack of rank n vector bundles on X. Let G denote the sheaf of automorphisms ofO_Xⁿ ⊕Ωⁿ preserving the symplectic form₂

(Oⁿ_X⊕Ωⁿ)→Ω. So, Gis a sheaf of groups in flat topology on the category ofX-schemes.

The stack Bun_GofG-bundles onX classifiesM ∈Bun_2ntogether with a symplectic form 2

M→Ω. Atheta-characteristicis a line bundleNonXequipped withN^⊗2→Ω. A choice of a theta-characteristic yields an isomorphismBun_G→Bun_Sp2n. So,Bun_Gis a smooth algebraic stack locally of finite type overk. SinceSp_2n is simply-connected, Bun_G is irreducible [11, Proposition 5]. Letd_G= dim Bun_G= (g−1) dimsp_2n. To express the dependence onnwe writeGn,BunGn,dGnand so on.

Denote by A the line bundle on BunG whose fibre at M is det RΓ(X, M) (cf. [8]). As χ(M) = 0, we viewAas aZ/2Z-graded line bundle placed in degree zero. It yields aμ2-gerbe

r:BunG→BunG. (10)

So,S-points ofBunG is the category: a line bundle B on S, a vector bundle M onS×X of rank 2n with symplectic form 2

M →Ω_S×X/S, and an isomorphism of OS-modules B²→det RΓ(X, M).

The idea of using the determinant of cohomology was communicated to me by G. Laumon and goes back to P. Deligne [9].

LetiBunG→BunGbe the locally closed substack given bydim H⁰(X, M) =i. LetiBunG

denote the preimage ofiBunGunderr.

LEMMA 1. – Each stratumiBunGofBunGis nonempty.

Proof. –Forn= 1 takeM =A(D)⊕(A^∗⊗Ω(−D)), where D is an effective divisor of degreeionX, andAis a line bundle onXof degreeg−1such thatH⁰(X,A) = H¹(X,A) = 0.

SuchAexist, becausedimX^(g−1)=g−1, and the dimension of the Picard scheme ofX isg.

Thendim H⁰(X, M) =i.

For any n constructM ∈iBun_G as M =M₁⊕ · · · ⊕M_n withM_j ∈ijBun_G1 for some i₁+· · ·+i_n=i. 2

We have a line bundleiBoniBunGwhose fibre atM∈BunGisdet H⁰(X, M). View it as a Z/2Z-graded placed in degreedim H⁰(X, M)modulo2. Then for eachiwe get aZ/2Z-graded isomorphism_iB²→ A|iBunG. By 3.1.2, the gerbe_iBun_G→iBun_G is trivial. So, we have the two-sheeted covering

iρ: Cov(_iBun_G)→iBun_G.

By [12, Theorem 17],Agenerates the Picard groupPic(Bun_G)→Z. So, the gerbris nontrivial, and the line bundles_iB(viewed as ungraded) do not glue into a line bundle overBun_G.

DEFINITION 1. – For eachidefine a local systemiAutoniBun_Gby

iAut = Hom_S2(sign,_iρ_!Q).

LetAutg∈P(BunG)(respectively,Auts∈P(BunG)) denote the Goresky–MacPherson exten- sion of 0Aut ⊗ Q[dG](^d₂^G) (respectively, of 1Aut ⊗ Q[dG − 1](^dG₂⁻¹)) under

iBunG→BunG.² Set

Aut = Aut_g⊕Aut_s. By construction,D(Aut)→Autcanonically.

Here is our main result.

THEOREM 1. – For eachithe∗-restrictionAut|

iBunGidentifies with Aut|iBunG→iAut⊗Q[1]

1 2

_⊗dG−i

(once √

−1∈k is fixed, the corresponding isomorphism is well-defined up to a sign). The

∗-restriction ofAutg(respectively, ofAuts)toiBunGvanishes foriodd(respectively, even).

Remark 1. – Classically, for two symplectic spacesW, W there is a natural mapSp(W )× Sp(W)→Sp(W⊕W), and the restriction of the metaplectic representation under this map is the tensor product of metaplectic representations of the factors [22, Remark 2.7].

In geometric setting we have a maps_n,m: Bun_Gn×Bun_Gm→Bun_Gn+msendingM, Mto M⊕M. It extends to a map

˜

sn,m:BunGn×BunGm→BunGn+m

2Here ‘g’ stands for generic and ‘s’ for special. We postpone to Proposition 7 the proof of the fact that1Autis a shifted perverse sheaf on1BunG.

e ◦

sending(M,B,B²→det RΓ(X, M))and(M,B,B²→det RΓ(X, M))to

M⊕M,B ⊗ B,B²⊗ B²→det RΓ(X, M)⊗det RΓ(X, M)→det RΓ(X, M⊕M) .

The restriction yields a mapsn,m:iBunGn×jBunGm→i+jBunGn+m and we get canonically s^∗_n,m(i+jB)→iBjB. For anyi, jthis yields an isomorphism

˜

s^∗_n,m(_i+jAut)→iAutjAut

of local systems oniBunGn×jBunGm. Thus,

˜

s^∗_n,mAut_g⊗Q[1]

1 2

_⊗dGn+dGm−dGn+m

→(Aut_gAut_g)⊕(Aut_sAut_s)

and

˜

s^∗_n,mAut_s⊗Q[1]

1 2

_⊗d_Gn+d_Gm−d_Gn+m

→(Aut_gAut_s)⊕(Aut_sAut_g) in the completed Grothendieck groupK(Bun_Gn×Bun_Gm)(the completion is with respect to the filtration given by the codimension of support).

3.3. For 1kn denote by Bun_Pk the stack classifying M ∈Bun_G together with an isotropic subbundle L₁⊂M of rank k. We write L₋₁⊂M for the orthogonal complement of L₁, so a point of Bun_Pk gives rise to a flag (L₁⊂L₋₁⊂M), and L₋₁/L₁∈Bun_Gn−k naturally.

Writeν_k: Bun_Pk→Bun_Gfor the projection. Define the map

˜

ν_k:Bun_Gn−k×Bun_Gn−kBun_Pk→Bun_G

as follows. AnS-point of the source is given by(L1⊂L₋1⊂M)∈BunPk(S)together with a (Z/2Z-graded of pure degree zero) invertibleOS-moduleBandB²→det RΓ(X, L₋1/L1). We have a canonical isomorphism ofZ/2Z-graded lines

det RΓ(X, L₁)⊗det RΓ(X, L₋₁/L₁)⊗det RΓ(X, L^∗₁⊗Ω)→det RΓ(X, M).

(11)

The map ν˜k sends this point to M ∈ BunG together with an invertible OS-module B=B ⊗det RΓ(X, L1) andB²→det RΓ(X, M)given by (11). SinceB is of pure degree asZ/2Z-graded, the map is well-defined by 3.1.2.

Let BunQk be the stack of collections: an exact sequence0→L1→L₋1→L₋1/L1→0 of vector bundles onX with L1∈Bunk and L₋1/L1∈Bun2n−2k, and a symplectic form 2

(L₋₁/L₁)→Ω(thus,L₋₁/L₁∈Bun_Gn−k).

Letη_k: Bun_Pk→Bun_Qk denote the natural projection. Let⁰Bun_Qk⊂Bun_Qk be the open substack given byH⁰(X,Sym²L₁) = 0.

THEOREM 2. – For the diagram

Bun_Gn−k×Bun_Gn−kBun_Qk←−−−−^id×ηk Bun_Gn−k×Bun_Gn−kBun_Pk−−→^˜νk Bun_G

we have an isomorphism

(id×η_k)_!ν˜^∗_kAut→AutQ[b]

b 2

overBunG_n−k×Bun_Gn−k0BunQk.(Once√

−1∈kis fixed, the isomorphism is well-defined up to a sign on generic and special parts.)Hereb(L1) =dG−dG_n−k−χ(L1)+2χ(Ω⁻¹⊗Sym²L1) is a function of a connected component of⁰BunQk. Ifχ(L1)is even then, over the corresponding connected component, the above isomorphism preserves generic and special parts, otherwise it interchanges them.

3.4. In Section 8.1 we consider the affine GrassmanianGrGforG, it is equipped with a natural line bundleLthat generates the Picard group ofGrG. LetGrG→GrGdenote theμ2-gerbe of square roots ofL. This is a local version of the gerbe (10). We introduce the categorySph(Gr_G) ofgenuine spherical sheavesonGr_G(cf. Definitions 4 and 6).

As for usual spherical sheaves on the affine Grassmanian, we equip Sph(GrG) with a structure of a rigid tensor category. Main result of Section 8 is the following version of the Satake equivalence.

THEOREM 3. – The categorySph(GrG) is canonically equivalent, as a tensor category, to the categoryRep(Sp_2n)of finite-dimensionalQ-representations ofSp_2n.

In Section 9 we define for K ∈ Sph(GrG) Hecke operators H(K,·) : D(BunG) → D(X ×BunG) compatible with the tensor structure on Sph(GrG). Finally, we prove Theo- rem 4 saying thatAutis a Hecke eigen-sheaf with eigenvalue

St = RΓ

P²ⁿ⁻¹,Q

⊗Q[1]

1 2

_⊗2n−1

viewed as a constant complex onX.

Remark 2. – The following observation was communicated to the author by Drinfeld. Letσbe the 2-automorphism ofid :Bun_G→Bun_Gthat acts on(M,₂

M→Ω,B²→det RΓ(X, M))∈ Bun_Gas−1onM and trivially onB. Thenσacts as−1onAut_sand trivially onAut_g. This is a way to think about the decomposition ofAutin a direct sum of perverse shaves.

4. Finite-dimensional model

4.1. LetV be a k-vector space of dimensiond. WriteST²(V^∗)for the space of symmetric tensors inV^∗⊗V^∗, this is the space of symmetric bilinear forms onV. Think ofb∈ST²(V^∗) as a mapb:V →V^∗such thatb^∗=b. Letp:V ×ST²(V^∗)→ST²(V^∗)denote the projection.

Letβ:V ×ST²(V^∗)→A¹be the map that sends(v, b)tov, bv. Set

Sψ=p!β^∗Lψ⊗Q[1]

1 2

_⊗d+¹₂d(d+1)

.

Letπ:V →Sym²V be the mapv→v⊗v. Then Sψ= Fourψ

π!Q[d]

d 2

. (12)

e ◦

The mapπis finite, andπ!Q=L0+L1, whereL0is the constant sheaf on the imageImπof π, andL1is a nontrivial local system of rank one onImπ− {0}extended by zero toImπ. So, Sψis a direct sum of two irreducible perverse sheaves.

LEMMA 2. –SψisGL(V)-equivariant.

Proof. –Clearly, π_!Q is GL(V)-equivariant. The Fourier transform preserves GL(V)- equivariance of a perverse sheaf. 2

StratifyST²(V^∗)byQi(V), whereQi(V)is the locus ofb:V →V^∗such thatdim Kerb=i.

Forb∈ST²(V^∗)denote byβ_b:V →A¹the map v→ v, bv. We have a usual ambiguity in identifyingST²(V^∗)withSym²(V^∗):bgoes toβ_bor ¹₂β_b. SinceS_ψisGL(V)-equivariant, we can view it as a perverse sheaf onSym²(V^∗)unambiguously.

LEMMA 3. – Forb∈Q0(V) the complex RΓc(V, β_b^∗Lψ) is a1-dimensional vector space placed in degreed.

Proof. –In some basisβb is given by (x1, . . . , xd)→x²₁+· · ·+x²_d. Thus we may assume d= 1. Consider the map π:A¹→A¹ given byπ(x) =x². As aboveπ!Q→ L0⊕ L1 with L0=Q. We getRΓc(A¹, π^∗Lψ)→RΓc(Gm,L⊗Lψ). The latter is a vector space of dimension one placed in degree one (a gamma-function onGm). 2

Let Cov(Q0(V))→Q0(V) denote the two-sheeted covering of Q0(V) whose fibre over b:V →V^∗is the set of trivializationsdetV →kwhose square is the one induced byb.

The groupGL(V)acts transitively onQ0(V), so givenb∈Q0(V)one gets an identification Q₀(V)→GL(V)/O(V, b). Our covering becomes the mapGL(V)/SO(V, b)→GL(V)/O(V).

More generally,GL(V)acts transitively onQ_i(V). Forb∈Q_i(V)withKerb=V₀, we can considerbas an element ofSym²(V /V₀)^∗. We get a parabolicP₀⊂GL(V)of automorphisms ofV that preserve V₀. LetSt_V0 be the preimage ofO(V /V0, b)underP₀→GL(V /V₀). Then StV0 is the stabilizer ofb∈Qi(V)inGL(V). SinceSO(V, b)is connected, for i < dthere is exactly one (up to isomorphism) nonconstantGL(V)-equivariant local system of rank one on Qi(V). It corresponds to theS2-coveringCov(Qi(V))→Qi(V)whose fibre overbis the set of trivializationsdet(V /V0)→kcompatible withb.

PROPOSITION 1. –

(1) The∗-restriction ofSψtoQi(V)is aGL(V)-equivariant local system of rank one placed in degreei−¹₂d(d+ 1). Fori < dthis local system is nonconstant and comes from the coveringCov(Qi(V))→Qi(V).

(2) Sψ=Sψ,g⊕Sψ,s is a direct sum of two irreducible perverse sheaves. Here Sψ,g is the Goresky–MacPherson extension ofS_ψ|Q0(V), andS_ψ,s is the Goresky–MacPherson extension ofS_ψ|Q1(V)underQ₁(V)→Q₁(V).

(3) We haveDS_ψ,g→S_ψ−1,gandDS_ψ,s→S_ψ−1,scanonically.

(4) If V =V1⊕V2 is a direct sum of two vector spaces of dimensions d1 andd2 then the

∗-restriction of Sψ⊗Q[1](¹₂)^{⊗−12d(d+1)} to the subspace Sym²(V₁^∗)⊕Sym²(V₂^∗) is canonically

(S_ψS_ψ)⊗Q[1]

1 2

_⊗−¹₂d1(d1+1)−¹2d2(d2+1)

.

Proof. –(2) A point ofQi(V)is given by a subspaceV0⊂V of dimensionitogether with non-degenerate formb:V /V0→(V /V0)^∗such thatb^∗=b. It follows that

dimQi(V) =1

2(d−i)(d+ 1−i) + (d−i)i=1

2(d−i)(d+ 1 +i).

From Lemma 3 applied toV /V0 we deduce thatSψ|Qi(V)is a local system of rank one placed in degreei−¹₂d(d+ 1). From (12) we see thatDS_ψ→S_ψ−1. For0idwe have

dimQi(V) =1

2(d−i)(d+ 1 +i)1

2d(d+ 1)−i,

the equality holds only fori= 0andi= 1. So,Sψis the Goresky–MacPherson extension from the open subschemeQ1(V).

LetSψ,gbe the intermediate extension ofSψ|Q0(V)toSym²V^∗. The∗-restrictionSψ,g|Q1(V)

vanishes. Indeed, it should be placed in strictly negative perverse degrees, but Sψ|Q1(V) is a perverse sheaf. Part (2) follows.

(3) Follows from (12).

(4) The compositionV₁⊕V₂→V −→^π Sym²V −→^a Sym²V₁×Sym²V₂ equalsπ×π. So, a_!π_!Q→(π_!Qπ_!Q). Fourier transform interchanges a_! and the ∗-restriction under the transpose mapa^∗: Sym²V₁^∗×Sym²V₂^∗→Sym²V^∗.

(1) SinceS_ψ|Qi(V)isGL(V)-equivariant, it remains to show it is nonconstant fori < d.

Step1. Start withd= 1case, soQ₀(V)→Gm. To show thatS_ψis nonconstant onQ₀(V)in this case, it suffices to prove thatRΓ_c(Gm, S_ψ) = 0.

We will show that RΓc(A¹×Gm, β^∗Lψ) = 0, where the map β:A¹×Gm→A¹ sends (v, b) to bv². Let β˜:A¹×Gm→A¹ be the map that sends (v, b) to bv. For the projection pr₁:A¹×Gm→A¹we have

pr_1!β˜^∗Lψ→j_∗Q[−1],

wherej:Gm→A¹is the open immersion [16, Lemma 2.3]. Letπ:A¹→A¹sendvtov². From the diagram

A¹×Gm π×id

pr₁

A¹×Gm β˜

pr₁

A¹

A^{1 π} A¹ we learn that

pr_1!β^∗Lψ→π^∗pr_1!β˜^∗Lψ.

It suffices to show that RΓ_c(A¹, π^∗j_∗Q) = 0. Recall that π_!Q→Q⊕ L1, where L1 is the local system on Gm extended by zero to A¹, which corresponds to the Galois covering π:Gm→Gm. We get

RΓc

A¹, π^∗j_∗Q

→RΓc

A¹, π!Q⊗j_∗Q

= 0, becauseRΓc(Gm,L1) = 0andRΓc(A¹, j_∗Q) = 0.

Step2. For anydandi < d choose a decomposition ofV as a direct sum V =W ⊕V1⊕

· · · ⊕Vd−i, wheredimVj= 1anddimW =i. ThenQ0(V1)× · · · ×Q0(Vd−i)⊂Qi(V). The restriction ofSψtoQ0(V1)× · · · ×Q0(Vd−i)is nonconstant by step 1 combined with (4). 2

PROPOSITION 2. – A choice of a square rooti=√

−1∈kyields for anyjan isomorphism

S_ψ⊗S_ψ|Qj(V)→Q[1]

1 2

⊗−2j+d(d+1)

.

e ◦

Proof. –Letβ2:V ×V ×Sym²V^∗→A¹be the map sending(v, u, b)tov, bv+u, bu.

Letp3:V ×V ×Sym²V^∗→Sym²V^∗be the projection. One checks that

S_ψ⊗S_ψ→p_3!β₂^∗Lψ⊗Q[1]

1 2

_⊗2d+d(d+1)

.

The change of variables

x=v+iu, y=v−iu

makes β2 to be the map sending(x, y, b) to x, by. Summate first over xwith y fixed, the assertion follows. 2

PROPOSITION 3. – The ∗-restriction Fourψ(Li)|Qj(V) vanishes if and only if j =i+ dmod 2. In other words, ifi=dmod 2thenFourψ(Li)has nontrivial fibres at

jevenQj(V).

Ifi=dmod 2thenFour_ψ(Li)has nontrivial fibres at

joddQ_j(V).

In particular, Fourψ(Li)[d](^d₂) =Sψ,g for i=d mod 2and Fourψ(Li)[d](^d₂) =Sψ,s for i=dmod 2.

Proof. –Ford= 1it is clear. Assume it is true ford−1.

The complexFour_ψ(Lj)isGL(V)-equivariant, andGL(V)acts transitively onQ_i(V). So, for eachiexactly one of two sheavesFour_ψ(L0)|Qi(V)orFour_ψ(L1)|Qi(V)vanishes, and the other is a rank one (shifted) local system.

Write V =V1⊕V2, where dimV1 =d−1 and dimV2= 1. Consider the natural map s: Sym²V →Sym²V1×Sym²V2. We have

s!(L0)→(L0L0)⊕(L1L1) and

s_!(L1)→(L0L1)⊕(L1L0), where on the right-hand sideLiare those forV₁andV₂.

Clearly,Qi−1(V1)×Q1(V2)→Qi(V)andQi(V1)×Q0(V2)→Qi(V). Consider Fourψ(L0)|Qi(V1)×Q0(V2)→h^∗

Fourψ(L0)Fourψ(L0) (13)

⊕h^∗

Four_ψ(L1)Four_ψ(L1) ,

whereh:Q_i(V₁)×Q₀(V₂)→Sym²V₁^∗×Sym²V₂^∗. This isomorphism holds up to a shift and a twist.

Ifi=d mod 2thenh^∗(Fourψ(L1)Fourψ(L1))is nonzero by induction hypothesis, so the LHS of (13) does not vanish, henceFourψ(L0)|Qi(V)does not vanish either.

If i=d mod 2 then the RHS of (13) vanishes by induction hypothesis, so the LHS also vanishes. Thus,Fourψ(L0)|Qi(V)vanishes. 2

4.2. Assume d1. Let Y(V) be the moduli scheme of pairs: a one-dimensional sub- space V₀⊂V and b ∈Sym²(V /V₀)^∗. The projection Y(V)→Gr(1, V) is a vector bun- dle, where Gr(1, V) denotes the Grassmanian of one-dimensional subspaces in V. Let α:Y(V)→Sym²V^∗be the map sending the above point to the composition

V →V /V0

−→b (V /V0)^∗→V^∗.