Geometric theta lifting for the dual pair SO2m Sp2n Sergey Lysenko

Description

Niveau: Supérieur, Doctorat, Bac+8
ar X iv :m at h/ 07 01 17 0v 2 [m ath .R T] 2 6 J un 20 07 Geometric theta-lifting for the dual pair SO2m, Sp2n Sergey Lysenko Abstract Let X be a smooth projective curve over an algebraically closed field of char- acteristic > 2. Consider the dual pair H = SO2m, G = Sp2n over X with H split. Write BunG and BunH for the stacks of G-torsors and H-torsors on X . The theta-kernel AutG,H on BunG?BunH yields the theta-lifting functors FG : D(BunH) ? D(BunG) and FH : D(BunG)? D(BunH) between the corresponding derived categories. We describe the rela- tion of these functors with Hecke operators. In two particular cases it becomes the geometric Langlands functoriality for this pair (in the nonramified case). Namely, we show that for n = m the functor FG : D(BunH) ? D(BunG) commutes with Hecke operators with respect to the inclusion of the Langlands dual groups Hˇ ?˜ SO2n ?? SO2n+1 ?˜ Gˇ. For m = n + 1 we show that the functor FH : D(BunG) ? D(BunH) commutes with Hecke operators with respect to the inclusion of the Langlands dual groups Gˇ ?˜ SO2n+1 ?? SO2n+2 ?˜ Hˇ .

• groups hˇ ?˜


• main local


• gˇ


• let xh ?


• module over


• so2n ??


• write pss


• so2n


• pair


• theta correspondence

Sujets

• Lafforgue
• Braverman
• Metaplectic group

Informations

• Publié par: profil-zyak-2012
• Nombre de lectures: 11
• Langue: English

Extrait

Geometric theta-lifting for the dual pair SO 2m Sp 2n

Sergey Lysenko

AbstractLetXprojective curve over an algebraically closed field of char-be a smooth acteristic> the dual pair2. ConsiderH= SO 2m , G=Sp 2n overXwithHsplit. Write Bun G and Bun H for the stacks ofG-torsors andH-torsors onX theta-kernel Aut. The GH on Bun G ×Bun H yields the theta-lifting functorsF G : D(Bun H )→D(Bun G ) andF H : D(Bun G )→D(Bun H ) between the corresponding derived categories. We describe the rela- tion of these functors with Hecke operators. In two particular cases it becomes the geometric Langlands functoriality for this pair (in the nonramified case). Namely, we show that forn=mthe functorF G : D(Bun H )→ D(Bun G ) commutes with Hecke operators with respect to the inclusion of the Langlands ˇ ˇ dual groupsHf→SO 2n ֒→SO 2n+1 →fG. Form=n+ 1 we show that the functorF H : D(Bun G )→D(Bun H with Hecke operators with respect to the inclusion of the) commutes ˇ ˇ Langlands dual groupsG→fSO 2n+1 ֒→SO 2n+2 →fH. In other cases the relation is more complicated and involves the SL 2 of Arthur. As a step of the proof, we establish the geometric theta-lifting for the dual pair GL m ,GL n  global. Our results are derived from the corresponding local ones, which provide a geometric analog of a theorem of Rallis.

1. Introduction

1.1 The classical Howe correspondence for dual reductive pairs is known to realize the Langlands functoriality in some particular cases (cf. [22], [1], [14]). In this paper, which is a continuation of [18], we depelop a similar geometric theory for dual reductive pairs (Sp 2n ,SO 2m ) and also (GL n ,GL m  consider only the everywhere unramified case.). We Remind the classical construction of the theta-lifting operators. LetXbe a smooth projective geometrically connected curve overF q . LetF=F q (X),Aesrid`elngofhtaeebX,Obe the integerad`eles.LetG,Hbe split connected reductive groups overF q that form a dual pair inside f some symplectic groupSp 2r . Assume that the metaplectic coveringSp 2r (A)→Sp 2r (A) splits overG(A)×H(A). LetSbe the corresponding Weil representation ofG(A)×H(A). A choice of ¯ a theta-functionalθ:S→Q ℓ yields a morphism of modules over the global nonramified Hecke algebrasH G ⊗ H H S (G×H)(O) →Funct((G×H)(F)\(G×H)(A)(G×H)(O)) sendingφto the function (g, h)7→θ((g, h)φ space). TheS (G×H)(O) has a distinguished nonram- ified vector, its imageφ 0  Viewingunder the above map is the classical theta-function.φ 0 as a kernel of integral operators, one gets the classical theta-lifting operators

F G : Funct(H(F)\H(A)H(O))→Funct(G(F)\G(A)G(O))

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and F H : Funct(G(F)\G(A)G(O))→Funct(H(F)\H(A)H(O)) For the dual pairs (Sp 2n ,SO 2m ) and (GL n ,GL m ) these operators realize the Langlands functo- riality between the corresponding automorphic representations (as we will see below, its precise formulation involves the SL 2  We establishof Arthur). a geometric analog of this phenomenon. Remind thatSf→ ⊗ ′ x∈X S x restricted tensor product of local Weil representations ofis the G(F x )×H(F x ). HereF x denotes the completion ofFatx∈X. The above functoriality in the x ) classical case is a consequence of a local result describing the space of invariantsS xG(O x )×H(O as a module over the tensor product x H G ⊗ x H H  the Inof local (nonramified) Hecke algebras. geomeric setting the main step is also to prove a local analog of this and then derive the global functoriality. Let us underline the following phenomenon in the proof that we find striking. LetG=Sp 2n , ˇ ˇ ¯ H= SO 2m . The Langlands dual groups areG→fSO 2n+1 andHf→SO 2m overQ ℓ . Write ˇ ˇ ¯ Rep(G) for the category of finite-dimensional representations ofGoverQ ℓ , and similarly for ˇ H. Roughly speaking, there will be algebraic varietiesY H , Y G overkand fully faithful functors ˇ ˇ f H : Rep(H)→P(Y H ) andf G : Rep(G)→P(Y G ) taking values in the categories of perverse sheaves (pure of weight zero) onY H (resp.,Y G ) with the following properties. Extendf H to a functor ˇ f H : Rep(H×G m )→ ⊕ i∈Z P(Y H )[i]⊂D(Y H ) ˇ naturally. That is, ifVis a representation ofHandIis the standard representation ofG m then f H (V⊠(I ⊗i ))f→f H (V)[i] is placed in perverse cohomological degree−i. Forn≥mthere will be a proper mapπ:Y G →Y H such that the following diagram is 2-commutative ˇ Rep(G) f → G P(Y G ) ↓ Res κ ↓ π ! ˇ f H ⊕ Rep(H×G m )→ i∈Z P(Y H )[i] ˇ ˇ for some homomoprhismκ:H×G m →G. Forn=mthe restriction ofκtoG m is trivial, soπ ! f G takes values in the category of perverse sheaves in this case. Bothf G andf H send an irreducible ˇ representation to an irreducible perverse sheaf. So, forV∈Rep(G) the decomposition of Res κ (V) into irreducible ones can be seen via the decomposition theorem of Beilinson, Bernstein and Deligne ([2]). To be more precise,Y H , Y G are not algebraic varieties, but ind-proobjects in the category of algebraic varieties. There will also be an analog of the above result forn < m (and also for the dual pair GL n ,GL m ). The above phenomenon is a part of our main local results (Proposition 4 in Section 5.1, Theorem 7 in Section 6.2). They provide a geometric analog of the local theta correspondence for these dual pairs. The key technical tools in the proof arethe weak geometric analogs of the Jacquet functors(cf. Section 4.7).

1.2 In the global setting let Ω denote the canonical line bundle onX. LetGbe the group scheme overXof automorphisms ofO X n ⊕Ω n preserving the natural symplectic form∧ 2 (O n X ⊕Ω n )→Ω.

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LetH= SO 2m  Bun. Write H for the stack ofH-torsors onX, similarly forG. Using the construction from [17], we introduce a geometric analog Aut GH of the above functionφ 0 , this is an object of the derived category ofℓ-adic sheaves on Bun G ×Bun H . It yields the theta-lifting functors F G : D(Bun H )→D(Bun G )

and F H : D(Bun G )→D(Bun H ) between the corresponding derived categories. Our mains global results for the pair (G, H) are Theorems 3 and 4 describing the relation between the theta-lifting functors and the Hecke functors on Bun G and Bun H  of the Oneagree with the conjectures of Adams ([1]). . They advantages of the geometric setting compared to the classical one is that the SL 2 of Arthur appears naturally (one can not even formulate a correct result in general without incorporating this SL 2 ). An essential difficulty in the proof was the fact that the complex Aut GH is not perverse (it has infinitely many perverse cohomologies), it is not even a direct sum of its perverse cohomologies (cf. Section 8.3.7). We also establish the global theta-lifting for the dual pair (GL n ,GL m ) (cf. Theorem 5). 1.3 Let us briefly duscuss how the paper is organized. Our main results are collected in Section 2. In Section 3 we remind some classical constructions at the level of functions, which we have in mind for geometrization. In Section 4 we develop a geometric theory for the following classical objects. LetK=F q ((t)) andO=F q [[t]]. Given a reductive groupGoverF q and its finite dimensional representation Mspace of invariants in the Schwarz space, the S(M(K)) G(O) is a module over the nonramified Hecke algebraH G analogs of the Fourier transform on this space introduce the geometric . We and (some weak analogs) of the Jacquet functors. A way to relate this with the global case is proposed in Section 4.6. In Section 5 we develop the local theta correspondence for the dual pair (GL n ,GL m ). The key ingredients here are decomposition theorem from [2], the dimension estimates from [19] and hyperbolic localization results from [3]. In Section 6 we develop the local theta correspondence for the dual pair (Sp 2n ,SO 2m ). In addition to the above tools, we use the classical result (Proposition 2) in the proof of our Propositions 7 and 8. In Section 7 we derive the global theta-lifting results for the dual pair (GL n ,GL m ). In Section 8 we prove our main global results (Theorems 3 and 4) about theta-lifting for the dual pair (Sp 2n ,SO 2m ). The relation between the local theory and the theta-kernel Aut GH comes from the results of [18]. In that paper we have introduced a schemeL d (M(F x )) of discrete lagrangian lattices in a symplectic Tate spaceM(F x ) and a certain 2 -gerb over it e e L d (M(F x )). The complex Aut GH on Bun GH comes from the stackL d (M(F x )) simply as the inverse image. The key observation is that it is much easier to prove the Hecke property of e Aut GH onL d (M(F x )), because over the latter stack it is perverse.

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In Section 8.3 we give another proof of a somewhat weaker statement than Theorem 4 using the ideas from [17]. The key step here is Proposition 12, which is partially inspired by the proof of the functional equation for geometric Eisenstein series ([4]). There seems no obvious way to derive Theorem 4 directly from Proposition 12, as Aut GH is not pure in general. However, at the level of functions or at the level of Grothendieck groups, Proposition 12 easily implies the classical analog of Theorem 4.

Acknowledgements.I am very grateful to Vincent Lafforgue for stimulating discussions which we have regularly for about last two years. They have contributed to this paper. He has also read the first version of the manuscript and indicated several mistakes. I also thank Alexander Braverman for nice discussions. Most of this work was realized in the University Paris 6 and the final part in the Institute for Advanced Study (Princeton), where the author was supported by NSF grant No. DMS-0111298.

2. Main results

2.1NotationFrom now onkdenotes an algebraically closed field of characteristicp >2 (except in Section 3, wherek=F q ). All the schemes (or stacks) we consider are defined overk. LetX Setbe a smooth projective connected curve.F=k(X). For a closed pointx∈X writeF x for the completion ofFatx, letO x ⊂F x be the ring of integers. LetD x = SpecO x denotes the disc aroundx. Write Ω for the canonical line bundle onX. Fix a primeℓ6=p a scheme (or stack). ForSwrite D(S) for the bounded derived category of ℓ´cida-onavesesheetalS, and P(S)⊂D(S DP() for the category of perverse sheaves. SetS) = ⊕ i∈Z P(S)[i]⊂D(S definition, we let for). ByK, K ′ ∈P(S), i, j∈Z Hom DP(S) (K[i], K ′ [j]) =  H 0, om P(S) (K, K ′ ),for fori i 6= = j j

Write P ss (S)⊂P(S DP Let) for the full subcategory of semi-simple perverse sheaves. ss (S)⊂ DP(Sbe the full subcategory of objects of the form) ⊕ i∈Z K i [i] withK i ∈P ss for alli. ¯ Fix a nontrivial characterψ:F p →Q ℓ ∗ and denote byL ψ the corresponding Artin-Shreier sheaf onA 1  we are working over an algebraically closed field, we systematically ignore. Since Tate twists. For a morphism of stacksf:Y→Zwe denote by dimrel(f) the function of a connected componentCofYgiven by dimC−dimC ′ , whereC ′ is the connected component of Zcontainingf(C). IfV→SandV ∗ →Sare dual rankrvector 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)[r], wherep V , p V ∗ are the projections, andξ:V× S V ∗ →A 1 is the pairing. Write Bun k for the stack of rankkvector bundles onX. Fork= 1 we also write PicX for the Picard stack Bun 1 ofX have a line bundle. WeA k on Bun k with fibre det RΓ(X, V) atV∈Bun k  it as a. ViewZ2Z-graded placed in degreeχ(V) mod 2. Our conventions about Z2Z-grading are those of ([17], 3.1).

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For a sheaf of groupsGon a schemeS,F 0 G denotes the trivialG-torsor onS. For a representationVofGand aG-torsorF G onSwe writeV F G =V× G F G for the induced vector bundle onS. In Section 8.2 we assume that the reader is familiar with the results of the first paper [18]

of this series. 2.2.1Hecke operatorsFor a connected reductive groupGoverk, letH G be the Hecke stack classifying (x,F G ,F ′ G , β), whereF G ,F ′ G areG-torsors onX,x∈Xandβ:F G | X−x →fF ′ G | X−x is an isomorphism. We have a diagram of projections X×Bun G supp ← ×h G ← H G h → G → Bun G , whereh G ← (resp.,h G → , supp) sends the above collection toF G (resp.,F ′ G ,x). Write x H G for the fibre ofH G overx∈X. ˇ LetT⊂B⊂Gbe a maximal torus and Borel subgroup, we write Λ G (resp., Λ G ) for the t Λ G + ˇ Λr(se + G  coweights) minant coweights (resp., weights) lattice ofG. Le p., denote the set of do (resp., dominant weights) ofG. Writeρˇ G (resp.,ρ G ) for the half sum of the positive roots (resp., ˇ ˇ coroots) ofG,w 0 the longest element of the Weyl group offor G. Forλ∈ Λˇ G + we writeV λ for the corresponding WeylG-module. Forx∈Xwe write Gr Gx for the affine grassmanianG(F x )G(O x ) (cf. [4], Section 3.2 for a detailed discussion). It can be seen as an ind-scheme classifying aG-torsorF G onXtogether with a trivializationβ:F G | X−x →fF G 0 | X−x overX−x. Forλ∈Λ G + let Gr x λ G ⊂Gr Gx be ˇ the closed subscheme classifying (F G , β) for whichV F 0 G (−hλ, λix)⊂V F G for everyG-moduleV whose weights are≤λ unique dense open ˇ. The G(O x )-orbit in Gr Gx λ is denoted Gr xG λ . ˇ LetA λ G denotes the intersetion cohomology sheaf of Gr λ G . LetGdenote the Langlands dual group toG. We write Sph G for the category ofG(O x )-equivariant perverse sheaves on Gr Gx . By ([19]), this is a tensor category, and there is a canonical equivalence of tensor categories ˇ ˇ ˇ  ¯ Loc : Rep(G)f→Sph G , where Rep(G) is the category ofG-representations overQ ℓ  this. Under equiv ceA λ G  ˇ corresponds to the irreducible repr alen esentation ofGwith h.w.λ. Write Bun Gx for the stack classifyingF G ∈Bun G together with a trivializationF G f→ F G 0 | D x . Following ([4], Section 3.2.4), write id l ,id r for the isomorphisms G(O x x H G →fBun Gx × ) Gr Gx such that the projection to the first factor corresponds toh G ← , h G → respectively. Let x H G λ ⊂ x H G be the closed substack that identifies with Bun Gx × G(O x ) Gr λ ia id l . Gx v ToS ∈Sph G , K∈D(Bun G ) one attaches their twisted external products (K ˜ ⊠S) l and ˜ (K⊠S) r on x H G , they are normalized to be perverse forK,Sperverse. The Hecke functors x H G ← , x H G → : Sph G ×D(Bun G )→D(Bun G )

are given by ˜ x H G ← (S, K) = (h G ← ) ! (∗S⊠K) r

and x H G → (S, K) = (h G → ) ! (S⊠ ˜ K) l

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We have denoted by∗: Sph G →fSph G the covariant equivalence of categories induced by the mapG(F x )→G(F x ),g7→g −1 Write also∗: Rep(G) ˇ →fRep(Gotrufcnidgnpsnoreorecthor)f ˇ . ˇ ˇ (in view of Loc), it sends an irreducibleG-module with h.w.λto the irreducibleG-module with h.w.−w 0 (λ). By ([6], Proposition 5.3.9), we have canonically x H G ← (∗S, K)f→ x H G → (S, K the). Besides, functorsK7→ x H G ← (S, K) andK7→ x H G → (D(S), K) are mutually (both left and right) adjoint. Lettingxmove aroundX, one similarly defines Hecke functors H G ← ,H G → : Sph G ×D(S×Bun G )→D(X×S×Bun G ), whereS Hecke functors are compatible with the tensor structure on Sph Theis a scheme. G and ˇ commute with Verdier duality (cf.loc.cit). Sometimes we write Rep(G) instead of Sph G in the definition of Hecke functors in view of Loc.

2.2.2 We introduce the category

D Sph G :=⊕ r∈Z Sph G [r]⊂D(Gr G ) It is equipped with a tensor structure, associativity and commutativity constraints so that ˇ the following holds. There is a canonical equivalence of tensor categories Loc r : Rep(G× G m )→fD Sph G such thatG m acts on Sph G [r] by the characterx7→x −r  the grading by. So, ˇ cohomological degrees in D Sph G corresponds to grading by the characters ofG m in Rep(G×G m ). In cohomological degree zero the equivalence Loc r specializes to Loc. The action of Sph G on D(Bun G  Sph) extends to an action of D G . Namely, we still denote by x H G ← , x H G → : D Sph G ×D(Bun G )→D(Bun G ) the functors given by x H G ← (S[r], K) = x H G ← (S, K)[r] and x H G → (S[r], K) = x H G → (S, K)[r] for S ∈Sph G andK∈D(Bun G ). We still denote by∗: D Sph G →D Sph G the functor given by∗(S[i]) = (∗S)[i] forS ∈Sph G . Write Loc X for the tensor category of local systems onX. Set

D Loc X =⊕ i∈Z Loc X [i]⊂D(X) We also equip it with a tensor structure so that a choice ofx∈Xyields an equivalence of tensor categories Rep(π 1 (X, x)×G m )→fD Loc X  Loc. The cohomological grading in D X corresponds to grading by the characters ofG m . For the standard definition of a Hecke eigen-sheaf we refer the reader to ([10], Section 2.7). Since we need to take into account the maximal torus of SL 2 of Arthur, we modify this standard definition as follows.

Definition 1.Given a tensor functorE: Sph G →D Loc X , aE-Hecke eigensheaf is a complex K∈D(Bun G ) equipped with an isomorphism H G ← (S, K)fE(S)⊠K[1] →

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functorial inS ∈Sph G satisfying the compatibility conditions (as inand loc.cit.). Note that oncex∈Xis chosen, a datum ofEbecomes equivalent to a datum of a homomorphism ˇ ˇ σ:π 1 (X, x)×G m →G. In other words, we are given a homomorphismG m →Gof algebraic ¯ groups overQ ℓ , and a continuous homomorphismπ 1 (X, x)→Z G ˇ (G m ), whereZ G ˇ (G m ) is the ˇ centralizer ofG m inG. ˇ ˇ Givenσ:π 1 (X, x)×G m →Gas above we writeσ ex :π 1 (X, x)×G m →G×G m for the homomorphism, whose first component isσ, and the second componentπ 1 (X, x)×G m →G m is the projection.

¯ Example1.The constant perverse sheafQ ℓ [dim Bun G ] on Bun G is aσ-Hecke eigensheaf for the ˇ ˇ homomoprhismσ:π 1 (X, x)×G m →Ggiven by 2ρ:G m →Gand trivial onπ 1 (X, x). 2.3Theta-sheafLetG r denote the sheaf of automorphisms ofO X r ⊕Ω r preserving the natural symplectic form∧ 2 (O r X ⊕Ω r )→ stack BunΩ. The G r ofG r -torsors onXsefiisclasM∈Bun 2r equipped with a symplectic form∧ 2 M→Ω. Remind the following objects introduced in [17]. WriteA G r for the line bundle on Bun G r with fibre det RΓ(X, M) atM view it as a. WeZ2Z Denote-graded line bundle purely of degree zero. g by Bun G r →Bun G r the 2 -gerbe of square roots ofA G r  theta-sheaf Aut = Aut. The g ⊕Aut s is g a perverse sheaf on Bun G r  for details).(cf. [17] 2.4.Dual pairSO 2m ,Sp 2n 2.4.1 Letn, m≥1,G=G n andA G =A G n . LetH= SO 2m be the split orthogonal group of rankmoverk. The stack Bun H ofH-torsors onXfies:assiclV∈Bun 2m , a nondegenerate symmetric form Sym 2 V→ O X , and a compatible trivializationγ: detV→ O X . LetA H be the f (Z2Z-graded) line bundle on Bun H  RΓ(with fibre detX, V) atV. Write Bun GH = Bun G ×Bun H . Let τ: Bun GH →Bun G 2nm be the map sending (M, V) toM⊗Vwith the induced symplectic form∧ 2 (M⊗V)→Ω. The following is proved in ([16], Proposition 2).

Lemma 1.There is a canonicalZ2Z-graded isomorphism of line bundles onBun GH τ ∗ A G 2nm f→ A 2Hn ⊗ A 2Gm ⊗det RΓ(X,O) −4nm (1) Letτ˜ : Bun GH →B g un G 2nm be the map sending (∧ 2 M→Ω,Sym 2 V→ O) to (∧ 2 (M⊗V)→ Ω,B), where B = det RΓ(X d , et V RΓ ) n ( ⊗ X, de O) tR 2n Γ m (X, M) m , andB 2 is identified with det RΓ(X, M⊗V) via (1).

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