Geometric theta lifting for the dual pair GSp2n GO2m Sergey Lysenko
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ar X iv :0 80 2. 04 57 v1 [ ma th. RT ] 4 F eb 20 08 Geometric theta-lifting for the dual pair GSp2n, GO2m Sergey Lysenko Abstract Let X be a smooth projective curve over an algebraically closed field of charac- teristic > 2. Consider the dual pair H = GO2m, G = GSp2n over X , where H splits over an etale two-sheeted covering π : X˜ ? X . Write BunG and BunH for the stacks of G-torsors and H-torsors on X . We show that for m ≤ n (respectively, for m > n) the theta-lifting functor FG : D(BunH) ? D(BunG) (respectively, FH : D(BunG) ? D(BunH)) commutes with Hecke functors with respect to a morphism of the corresponding L-groups involving the SL2 of Arthur. In two particular cases n = m and m = n + 1 this becomes the geometric Langlands functoriality for the corresponding dual pair. As an application, we prove a particular case of the geometric Langlands conjectures. Namely, we construct the automorphic Hecke eigensheaves on BunGSp4 corresponding to the endoscopic local systems on X . 1. Introduction 1.1 The classical theta correspondence for the dual reductive pair (GSp2n,GO2m) is known to satisfy a version of strong Howe duality (cf.
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Publié par
Langue
English
Geometric theta-lifting for
the dual pair GSp2n, GO2m
Sergey Lysenko
AbstractLetXbe a smooth projective curve over an algebraically closed field of charac-teristic>2. Consider the dual pairH= GO2m,G= GSp2noverX, whereHsplits over an ˜ e´taletwo-sheetedcoveringπ:X→X Bun. WriteGand BunHfor the stacks ofG-torsors andH-torsors onX show that for. Wem≤n(respectively, form > n) the theta-lifting functorFG: D(BunH)→D(BunG) (respectively,FH: D(BunG)→D(BunH)) commutes with Hecke functors with respect to a morphism of the corresponding L-groups involving the SL2 two particular cases Inof Arthur.n=mandm=n+ 1 this becomes the geometric Langlands functoriality for the corresponding dual pair. As an application, we prove a particular case of the geometric Langlands conjectures. Namely, we construct the automorphic Hecke eigensheaves on BunGSp4corresponding to the endoscopic local systems onX.
1. Introduction
1.1 The classical theta correspondence for the dual reductive pair (GSp2n,GO2m) is known to satisfy a version of strong Howe duality (cf. [12]). In this paper, which is a continuation of [7], we develop the geometric theory of theta-lifting for this dual pair in the everywhere unramified case. The classical theta-lifting operators for this dual pair are as follows. LetXbe a smooth projective geometrically connected curve overFq(withqodd). LetF=Fq(X),Aele`sbethead ring ofX,OΩeofrWtiacontrehllinnicadleoebunnhtnteiereg`eads.leX a rank. Pick 2n-vector bundleMwith symplectic form∧2M→ Awith values in a line bundleAonX. Let Gbe the group scheme overXof automorphisms of the GSp2n-torsor (M,A). ˜ Letπ:X→XngrithwiloGagrisowteehs-deteevoctelaae´nbuoΣp={1, σ}. LetE be theσ-anti-invariants inπ∗OX˜ a rank 2. Fixm-vector bundleVonXwith symmetric form Sym2V→ Cwith values in a line bundleConXtogether with a compatible trivialization γ:C−m⊗detVf→ E. This means thatγ2:C−2m⊗(detV)2→ Ois the trivialization induced f ˜ by the symmetric form. LetHbe the group scheme overXof automorphisms ofVpreserving the symmetric form up to a multiple and fixingγ is a form of G. ThisO02m, where GO20mis the connected component of unity of the split orthogonal similitude group. Assume given an isomorphismA ⊗ Cf→Ω.
LetG2nmthe group scheme of automorphisms ofM⊗Vpreserving the symplectic form 2 ˜( ˜ ∧M⊗V)→Ω. WriteGH⊂G×Hfor the group subscheme overXof pairs (g, h) such e thatg⊗hacts trivially onA ⊗ C. The metaplectic coverG2nm(A)→G2nm(A) splits naturally ˜ after restriction underGH(A)→G2nm(A). LetSbe the corresponding Weil representation
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˜ ofGH(˜A). The spaceSGH(O)has a distinguished nonramified vectorv0. Ifθ:S→Q¯ℓis a ˜ ˜ ˜ ¯ theta-functional thenφ0:GH(F)\GH(A)GH(O)→Qℓgiven byφ0(g, h) =θ((g, h)v0) is the classical theta-function. The theta-lifting operators ˜ ˜ ˜ FG: Funct(H(F)\H(A)H(O))→Funct(G(F)\G(A)G(O))
and ˜ ˜ ˜ FH˜: Funct(G(F)\G(A)G(O))→Funct(H(F)\H(A)H(O)) are the integral operators with kernelφ0for the diagram of projections ˜ ˜ ˜ GH(F)\GH(A)GH(O) ւqցp ˜ ˜ ˜ H(F)\H(A)H(O)G(F)\G(A)G(O)
The following statement would be an analog of a theorem of Rallis [11] for similitude groups (the author have not found its proof in the litterature). Ifm≤n(resp.,m > n) thenFG (resp.,FH˜commutes with the actions of global Hecke algebras) HG,HH˜with respect to certain homomorphismHG→ HH˜(resp.,HH˜→ HG ). Weprove a geometric version of this result (cf. Theorem 1). Its precise formulation in the geometric setting involves the SL2of Arthur (or rather its maximal torus). In the particular casen=m(resp.,m=n+ 1) the SL2of Arthur dissapears, and the corresponding morphisms of Hecke algebras come from morphisms of L-groupsHL→GL(resp,GL→HL). Our methods extend those of [7], the global results are derived from the corresponding local ones. Remind thatSf→ ⊗′x∈XSxis the restricted tensor product of local Weil representations. LetFxbe the completion ofFatx∈X,Ox⊂Fx Thethe ring of integers. geometric ˜e analog of theGH(Fxrpse-)eritnoneatSxis the Weil categoryW(Ld(W0(Fx Sections))) (cf. 3.1-3.2). Informally speaking, we work rather with the geometric analog of the compactly induced representation ˜ Sin ¯x d= c-(GHG˜×(FHx)()Fx)Sx e Its manifestation is a family of categories DTa(Ld(Wa(Fx))) indexed bya∈Z(cf. Section 4.2). Our main local result is Theorem 3. In classical terms, it compares the action of Hecke ˜ ¯ operators forGandHon the natural nonramified vector inSx a byproduct, we also obtain. As some new results at the classical level of functions (Propositions A.1 and A.2). Foraeven they reduce to a result from [10], but foraare new and amount to a calculationodd they ofK×SO(Ox)-invariants in the Weil representation of (Sp2n×SO2m)(Fx), whereKis the nonstandard maximal compact subgroup ofSp2n(Fx). 1.2 The most striking application of our Theorem 1 is a proof of the following particular case of the geometric Langlands conjecture forG= GSp4. LetEbe an irreducible rank 2 smooth ¯ ˜ ¯ Qℓ-sheaf onXequipped with an isomorphismπ∗χf→detE, whereχis a smoothQℓ-sheaf on Xof rank one. Thenπ∗(E∗equipped with a natural symplectic form) is ∧2(π∗E∗)→χ−1, so ˇ ˇ ¯ can be viewed as aG-local systemEGˇonX, whereGis the Langlands dual group overQℓ. We
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construct the automorphic sheafKon BunG, which is a Hecke eigensheaf with respect toEGˇ (cf. Corollary 1). Acknowledgements.I am grateful to V. Lafforgue for regular and stimulating discussions.
2. Main results
2.1NotationFrom now onkdenotes an algebraically closed field of characteristicp >2, all the schemes (or stacks) we consider are defined overk(except in Section 4.8.7.2). Fix a primeℓ6=p. For a scheme (or stack)Swrite D(S) for the bounded derived category of ℓa-id´ctelaseehesavonS, and P(S)⊂D(S Set) for the category of perverse sheaves. DP(S) = ⊕i∈ZP(S)[i]⊂D(S definition, we let for). ByK, K′∈P(S), i, j∈Z omP(S)(K HomDP(S)(K[i], K′[j]) =0H,,K′),forforii6==jj Since we are working over an algebraically closed field, we systematically ignore Tate twists (except in Section 4.8.7.2, where we work over a finite subfieldk0⊂k this case we also fix a. In ¯ ¯ ¯ square rootQℓ(2) of the sheafQℓ(1) over Speck0 a nontrivial character). Fixψ:Fp→Qℓ∗and denote byLψthe corresponding Artin-Shreier sheaf onA1. IfV→SandV∗→Sare dual ranknvector bundles over a stackS, we normalize the Fourier transform Fourψ: D(V)→D(V∗) by Fourψ(K) = (pV∗)!(ξ∗Lψ⊗p∗VK)[n](2), where pV, pV∗are the projections, andξ:V×SV∗→A1is the pairing. For a sheaf of groupsGon a schemeS,F0Gdenotes the trivialG-torsor onS a. For representationVofGand aG-torsorFGonSwriteVFG=V×GFGfor the induced vector bundle onS. For a morphism of stacksf:Y→Zdenote by dimrel(f) the function of connected componentCofYgiven by dimC−dimC′, whereC′is the connected component of Zcontainingf(C).
2.2Hecke operatorsLetXbe a smooth connected projective curve. Forr≥1 write Bunr for the stack of rankrvector bundles onX Picard stack Bun. The1is also denoted PicX. For a connected reductive groupGoverk, let BunGdenote the stack ofG-torsors onX. ˇ Given a maximal torus and a Borel subgroupT⊂B⊂G, we write ΛG(resp., ΛG) for the coweights (resp., weights) lattice ofG Λ. LetG+ˇΛ,.pser(+G) denote the set of dominant coweights (resp., dominant weights) ofG. WriteρˇG(resp.,ρG) for the half sum of the positive roots (resp., coroots) ofG,w0for the longuest element of the Weyl group ofG. SetK=k(X). For a closed pointx∈XletKxbe the completion ofKatx,Ox⊂Kxbe its ring of integers. The following notations are borrowed from [7]. Write GrGxfor the affine grassmanian G(Kx)G(Ox is an ind-scheme classifying a). ThisG-torsorFGonXtogether with a trivial-izationβ:FG|X−xf→ F0G|X−x. Forλ∈ΛG+write GrλGx⊂GrGxfor the closed subscheme ˇ ˇ classifying (FG, β) for whichVFG0(−hλ, λix)⊂VFGfor everyG-moduleVwhose weights are≤λ. The unique dense openG(Ox)-orbit in GrλGxis denoted GrλGx.
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Forθ∈π1(G) the connected component GrθGof GrGclassifies pairs (FG, β) such that ˇ ˇ VFG0(−hθ, λi)→fVFGfor every one-dimensionalG-module with highest weightλ. Denote byAλG fthe intersection coho oλˇ mology shea f GrG. WriteGfor the Langlands dual ¯ group toG, this is a reductive group overQℓequipped with the dual maximal torus and Borel ˇ ˇ ˇ subgroupT⊂B⊂G Sph. WriteGfor the category ofG(Ox)-equivariant perverse sheaves on GrGx. This is a tensor category, and one has a canonical equivalence of tensor categories ˇ ˇ ˇ Loc : Rep(G)→fSphG, where Rep(G) is the category of finite-dimensional representations ofG ¯ overQℓ(cf. [9]). For the definition of the Hecke functors HG←,HG→: SphG×D(BunG)→D(X×BunG) we refer the reader to ([7], Section 2.2.1). Write∗: SphGf→SphGfor the covariant equivalence induced by the mapG(Kx)→G(Kx),g7→g−1. In view of Loc, the corresponding functor ˇ ˇ ˇ ˇ ∗: Rep(G)f→Rep(G) sends an irreducibleG-module with h.w.λto the irreducibleG-module with h.w.−w(λ). Forλ∈Λ+we also write Hλ() = H←(Aλ,).
0 G GG G Set D SphG=⊕r∈ZSphG[r]⊂D(GrG) As in ([7], Section 2.2.2), we equip it with a structure of a tensor category in such a way that the ˇ Satake equivalence extends to an equivalence of tensor categories Locr: Rep(G×Gm)f→D SphG. Our convention is thatGmacts on SphG[r] by the characterx7→x−r. ˜ Now letπ:X→X-omohaneviG.ΣpougrisloGathwingebe´etinfialoGaletarivecois morphism Σ→Aut(G), letGbe the group scheme onXobtained as the twisting ofGby the ˜ ˜ ˜ ˜ Σ-torsorπ:X→X. SetK=k(X a closed point). Forx˜∈XwriteK˜xfor the completion of ˜ Kat ˜x,O˜⊂K˜xfor its ring of integers, and GrG˜xfor the affine grassmanianG(K˜x)G(O˜x). x Write BunGfor the stack ofG-torsors onX. One defines Hecke functors x˜HG←,x˜H→SphG×D(BunG)→D(BunG) (1) G: as follows. Write˜xHGfor the Hecke stack classifyingG-torsorsFG,F′GonXand an isomorphism FG→fF′G|X−π(x˜). We have a diagram BunhG←←x˜HhG→→BunG, whereh←(resp.,h→) sends (FG,F′G,˜x) toFG(resp., toF′G). SetD˜x= SpecO˜x Bun. LetGx˜ 0 be the stack classifyingFG∈BunGtogether with a trivializationFG|Dx˜f→ FG. Write idl,idr for the isomorphisms x˜HG→fBunGx˜×G(Ox˜)GrGx˜ such that the projection to the first factor corresponds toh←, h→respectively. ToS ∈SphG, K∈D(BunG) one attaches their twisted external product (K˜⊠S)land (K⊠˜S)ronx˜HG, they are normalized to be perverse forK, S functors (1) are defined byperverse. The ) =h!←(K⊠˜∗ S)randx˜HG→(S, K) =h!→(K˜l ˜xHG←(SK⊠S) ,
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