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: pefav
• Nombre de lectures: 8
• Langue: English

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Geometric theta-lifting for

the dual pair GSp 2n , GO 2m

Sergey Lysenko

AbstractLetXbe a smooth projective curve over an algebraically closed field of charac- teristic>2. Consider the dual pairH= GO 2m ,G= GSp 2n overX, whereHsplits over an ˜ e´taletwo-sheetedcoveringπ:X→X Bun. Write G and Bun H for the stacks ofG-torsors andH-torsors onX show that for. Wem≤n(respectively, form > n) the theta-lifting functorF G : D(Bun H )→D(Bun G ) (respectively,F H : D(Bun G )→D(Bun H )) commutes with Hecke functors with respect to a morphism of the corresponding L-groups involving the SL 2  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 Bun GSp 4 corresponding to the endoscopic local systems onX.

1. Introduction

1.1 The classical theta correspondence for the dual reductive pair (GSp 2n ,GO 2m ) 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 overF q (withqodd). LetF=F q (X),Aele`sbethead ring ofX,OΩeofrWtiacontrehllinnicadleoebunnhtnteiereg`eads.leX a rank. Pick 2n-vector bundleMwith symplectic form∧ 2 M→ Awith values in a line bundleAonX. Let Gbe the group scheme overXof automorphisms of the GSp 2n -torsor (M,A). ˜ Letπ:X→XngrithwiloGagrisowteehs-deteevoctelaae´nbuoΣp={1, σ}. LetE be theσ-anti-invariants inπ ∗ O X ˜  a rank 2. Fixm-vector bundleVonXwith symmetric form Sym 2 V→ Cwith values in a line bundleConXtogether with a compatible trivialization γ:C −m ⊗detVf→ E. This means thatγ 2 :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. ThisO 0 2m , where GO 20 m is the connected component of unity of the split orthogonal similitude group. Assume given an isomorphismA ⊗ Cf→Ω.

LetG 2nm the 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 coverG 2nm (A)→G 2nm (A) splits naturally ˜ after restriction underGH(A)→G 2nm (A). LetSbe the corresponding Weil representation

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˜ ofGH( ˜ A). The spaceS GH(O) has a distinguished nonramified vectorv 0 . Ifθ:S→Q ¯ ℓ is a ˜ ˜ ˜ ¯ theta-functional thenφ 0 :GH(F)\GH(A)GH(O)→Q ℓ given byφ 0 (g, h) =θ((g, h)v 0 ) is the classical theta-function. The theta-lifting operators ˜ ˜ ˜ F G : Funct(H(F)\H(A)H(O))→Funct(G(F)\G(A)G(O))

and ˜ ˜ ˜ F H ˜ : Funct(G(F)\G(A)G(O))→Funct(H(F)\H(A)H(O)) are the integral operators with kernelφ 0 for the diagram of projections ˜ ˜ ˜ GH(F)\GH(A)GH(O) ւ q ց p ˜ ˜ ˜ 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) thenF G (resp.,F H ˜ commutes with the actions of global Hecke algebras) H G ,H H ˜ with respect to certain homomorphismH G → H H ˜ (resp.,H H ˜ → H G  ). Weprove a geometric version of this result (cf. Theorem 1). Its precise formulation in the geometric setting involves the SL 2 of Arthur (or rather its maximal torus). In the particular casen=m(resp.,m=n+ 1) the SL 2 of Arthur dissapears, and the corresponding morphisms of Hecke algebras come from morphisms of L-groupsH L →G L (resp,G L →H L ). Our methods extend those of [7], the global results are derived from the corresponding local ones. Remind thatSf→ ⊗ ′ x∈X S x is the restricted tensor product of local Weil representations. LetF x be the completion ofFatx∈X,O x ⊂F x  Thethe ring of integers. geometric ˜e analog of theGH(F x rpse-)eritnoneatS x is the Weil categoryW(L d (W 0 (F x  Sections))) (cf. 3.1- 3.2). Informally speaking, we work rather with the geometric analog of the compactly induced representation ˜ Sin ¯ x  d= c- ( GH G ˜ × (F H x ) ()F x ) S x e Its manifestation is a family of categories D T a (L d (W a (F x ))) 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 inS x  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(O x )-invariants in the Weil representation of (Sp 2n ×SO 2m )(F x ), whereKis the nonstandard maximal compact subgroup ofSp 2n (F x ). 1.2 The most striking application of our Theorem 1 is a proof of the following particular case of the geometric Langlands conjecture forG= GSp 4 . LetEbe an irreducible rank 2 smooth ¯ ˜ ¯ Q ℓ -sheaf onXequipped with an isomorphismπ ∗ χf→detE, whereχis a smoothQ ℓ -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 systemE G ˇ onX, whereGis the Langlands dual group overQ ℓ . We

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construct the automorphic sheafKon Bun G , which is a Hecke eigensheaf with respect toE G ˇ (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∈Z P(S)[i]⊂D(S definition, we let for). ByK, K ′ ∈P(S), i, j∈Z om P(S) (K Hom DP(S) (K[i], K ′ [j]) =  0 H , ,K ′ ), for fori i6 = = j j 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 subfieldk 0 ⊂k this case we also fix a. In ¯ ¯ ¯ square rootQ ℓ ( 2 ) of the sheafQ ℓ (1) over Speck 0  a nontrivial character). Fixψ:F p →Q ℓ ∗ and denote byL ψ the corresponding Artin-Shreier sheaf onA 1 . IfV→SandV ∗ →Sare dual ranknvector bundles over a stackS, we normalize the Fourier transform Four ψ : D(V)→D(V ∗ ) by Four ψ (K) = (p V ∗ ) ! (ξ ∗ L ψ ⊗p ∗ V K)[n]( 2 ), where p V , p V ∗ are the projections, andξ:V× S V ∗ →A 1 is the pairing. For a sheaf of groupsGon a schemeS,F 0 G denotes the trivialG-torsor onS a. For representationVofGand aG-torsorF G onSwriteV F G =V× G F G for the induced vector bundle onS. For a morphism of stacksf:Y→Zdenote by dimrel(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 Bun r for the stack of rankrvector bundles onX Picard stack Bun. The 1 is also denoted PicX. For a connected reductive groupGoverk, let Bun G denote the stack ofG-torsors onX. ˇ Given a maximal torus and a Borel subgroupT⊂B⊂G, we write Λ G (resp., Λ G ) for the coweights (resp., weights) lattice ofG Λ. Let G + ˇ Λ,.pser( + G ) denote the set of dominant coweights (resp., dominant weights) ofG. Writeρˇ G (resp.,ρ G ) for the half sum of the positive roots (resp., coroots) ofG,w 0 for the longuest element of the Weyl group ofG. SetK=k(X). For a closed pointx∈XletK x be the completion ofKatx,O x ⊂K x be its ring of integers. The following notations are borrowed from [7]. Write Gr Gx for the affine grassmanian G(K x )G(O x  is an ind-scheme classifying a). ThisG-torsorF G onXtogether with a trivial- izationβ:F G | X−x f→ F 0 G | X−x . Forλ∈Λ G + write Gr λ Gx ⊂Gr Gx for the closed subscheme ˇ ˇ classifying (F G , β) for whichV F G 0 (−hλ, λix)⊂V F G for everyG-moduleVwhose weights are≤λ. The unique dense openG(O x )-orbit in Gr λ Gx is denoted Gr λ Gx .

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Forθ∈π 1 (G) the connected component Gr θ G of Gr G classifies pairs (F G , β) such that ˇ ˇ V F G 0 (−hθ, λi)→fV F G for every one-dimensionalG-module with highest weightλ. Denote byA λ G  fthe intersection coho o λ ˇ mology shea f Gr G . WriteGfor the Langlands dual ¯ group toG, this is a reductive group overQ ℓ equipped with the dual maximal torus and Borel ˇ ˇ ˇ subgroupT⊂B⊂G Sph. Write G for the category ofG(O x )-equivariant perverse sheaves on Gr Gx . This is a tensor category, and one has a canonical equivalence of tensor categories ˇ ˇ ˇ Loc : Rep(G)→fSph G , where Rep(G) is the category of finite-dimensional representations ofG ¯ overQ ℓ (cf. [9]). For the definition of the Hecke functors H G ← ,H G → : Sph G ×D(Bun G )→D(X×Bun G ) we refer the reader to ([7], Section 2.2.1). Write∗: Sph G f→Sph G for the covariant equivalence induced by the mapG(K x )→G(K x ),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 H λ () = H ← (A λ ,).

0  G  GG G Set D Sph G =⊕ r∈Z Sph G [r]⊂D(Gr G ) 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 Loc r : Rep(G×G m )f→D Sph G . Our convention is thatG m acts on Sph G [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 ˜x for the completion of ˜ Kat ˜x,O ˜ ⊂K ˜x for its ring of integers, and Gr G˜x for the affine grassmanianG(K ˜x )G(O ˜x ). x Write Bun G for the stack ofG-torsors onX. One defines Hecke functors x˜ H G ← , x˜ H → Sph G ×D(Bun G )→D(Bun G ) (1) G : as follows. Write ˜x H G for the Hecke stack classifyingG-torsorsF G ,F ′ G onXand an isomorphism F G →fF ′ G | X−π(x˜) . We have a diagram Bun h G ← ← x˜ H h G → → Bun G , whereh ← (resp.,h → ) sends (F G ,F ′ G ,˜x) toF G (resp., toF ′ G ). SetD ˜x = SpecO ˜x  Bun. Let Gx˜ 0 be the stack classifyingF G ∈Bun G together with a trivializationF G | D x˜ f→ F G . Write id l ,id r for the isomorphisms x˜ H G →fBun Gx˜ × G(O x˜ ) Gr Gx˜ such that the projection to the first factor corresponds toh ← , h → respectively. ToS ∈Sph G , K∈D(Bun G ) one attaches their twisted external product (K ˜ ⊠S) l and (K⊠ ˜ S) r on x˜ H G , they are normalized to be perverse forK, S functors (1) are defined byperverse. The ) =h ! ← (K⊠ ˜ ∗ S) r and x˜ H G → (S, K) =h ! → (K ˜ l ˜x H G ← (SK⊠S) ,

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