arXiv subjects

Sergey Lysenko

Publications and source records attributed to Sergey Lysenko.

26 records · Page 2Linked to original sources

arXiv2007

Geometric Weil representation: local field case

Let k be an algebraically closed field of characteristic >2, F=k((t)) and Mp(F) denote the metaplectic extension of Sp_{2d}(F). In this paper we propose a geometric analog of the Weil representation of Mp(F). This is a category of certain perverse sheaves on some stack, on which Mp(F) acts by functors. This construction will be used in math.RT/0701170 (and subsequent publications) for a proof of the geometric Langlands functoriality for some dual reductive pairs.

math.RT↗

arXiv2007

Geometric Waldspurger periods

This paper is a step towards a version of the theta-lifting (or Howe correspondence) in the framework of the geometric Langlands program. We consider the (unramified) dual reductive pair H=GO_{2m}, G=GSp_{2n} over a smooth projective curve X (our H is assumed to be split on an etale two-sheeted covering Y of X). Let Bun_G and Bun_H be the stack of G-torsors and H-torsors on X. We define and study functors F_G and F_H between the derived categories D(Bun_G) and D(Bun_H) that are analogs of the classical theta-lifting operators. One of the main results is the geometric Langlands functoriality for the dual pair (H=GO_2, G=GL_2), where GO_2 is the direct image of the multiplicative group from a nontrivial etale covering Y to X. The functor F_G from D(Bun_H) to D(Bun_G) commutes with Hecke operators with respect to the corresponding map of Langlands L-groups G^L\to H^L. As an application, we calculate Waldspurger periods of cuspidal automorphic sheaves on Bun_{GL_2} and Bessel periods of theta-lifts from Bun_{GO_4} to Bun_{GSp_4}. Based on these calculations, we give three conjectural constructions of certain automorphic sheaves on Bun_{GSp_4} (one of them makes sense for D-modules only).

math.AG↗

arXiv2005

Moduli of metaplectic bundles on curves and Theta-sheaves

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 curve X we introduce an algebraic stack \tilde\Bun_G of metaplectic bundles on X. It also has a local version \tilde\Gr_G, which is a gerbe over the affine grassmanian of G. We define a categorical version of the (nonramified) Hecke algebra of the metaplectic group. This is a category Sph(\tilde\Gr_G) of certain perverse sheaves on \tilde\Gr_G, which act on \tilde\Bun_G by Hecke operators. A version of the Satake equivalence is proved describing Sph(\tilde\Gr_G) as a tensor category. Further, we construct a perverse sheaf on \tilde\Bun_G corresponding to the Weil representation and show that it is a Hecke eigen-sheaf.

math.AG↗

arXiv2005

Geometric Bessel models for GSp_4 and multiplicity one

I this paper, which is a sequel to math.AG/0310361, we study Bessel models of representations of GSp_4 over a local non archimedian field in the framework of the geometric Langlands program. The Bessel module over the nonramified Hecke algebra of GSp_4 admits a geometric counterpart, the Bessel category of perverse sheaves on some ind-algebraic stack. We use it to prove a geometric version of the multiplicity one for Bessel models. It implies a geometric Casselman-Shalika type formula for these models. The strategy of the proof is the same as in the paper of Frenkel, Gaitsgory and Vilonen math.AG/9907133. We also propose a geometric framework unifying Whittaker, Waldspurger and Bessel models.

math.AG↗

arXiv2005

Whittaker and Bessel functors for GSp_4

One of the important technical tools in Gaitsgory's proof of the Vanishing Conjecture appearing in the geometric Langlands correspondence ([3]) is the theory of Whittaker functors for GL_n. We define Whittaker functors for GSp_4 and study their properties. In a sense, these functors correspond to the maximal parabolic subgroup of GSp_4, whose unipotent radical is not commutative. We also study similar functors corresponding to the Siegel parabolic subgroup of GSp_4, they are related with Bessel models for GSp_4 and Waldspurger models for GL_2. We define the Waldspurger category, which is a geometric counterpart of the Waldspurger module over the Hecke algebra of GL_2. We prove a geometric version of the multiplicity one result for Waldspurger models.

math.AG↗

arXiv2003

On automorphic sheaves on Bun_G

Let X be a smooth projective connected curve over an algebraically closed field k of positive characteristic. Let G be a reductive group over k, γbe a dominant coweight for G, and E be an \ell-adic \check{G}-local system on X, where \check{G} denotes the Langlands dual group. Let \Bun_G be the moduli stack of G-bundles on X. Under some conditions on the triple (G,γ,E) we propose a conjectural construction of a distinguished E-Hecke automorphic sheaf on \Bun_G. We are motivated by a construction of automorphic forms suggested by Ginzburg, Rallis and Soudry in [6,7]. We also generalize Laumon's theorem ([10], Theorem 4.1) for our setting. Finally, we formulate an analog of the Vanishing Conjecture of Frenkel, Gaitsgory and Vilonen for Levi subgroups of G.

math.RT↗

arXiv2001

Global geometrised Rankin-Selberg method for GL(n)

We propose a geometric interpretation of the classical Rankin-Selberg method for GL(n) in the framework of the geometric Langlands program. We show that the geometric Langlands conjecture for an irreducible unramified local system $E$ of rank $n$ on a curve implies the existence of automorphic sheaves corresponding to the universal deformation of $E$. Then we calculate the `scalar product' of two automorphic sheaves attached to this universal deformation.

math.AG↗

arXiv2001

Local geometrised Rankin-Selberg method for GL(n)

Following Laumon [10], to a nonramified $\ell$-adic local system $E$ of rank $n$ on a curve $X$ one associates a complex of $\ell$-adic sheaves $_n{\cal K}_E$ on the moduli stack of rank $n$ vector bundles on $X$ with a section, which is cuspidal and satisfies Hecke property for $E$. This is a geometric counterpart of the well-known construction due to Shalika [17] and Piatetski-Shapiro [16]. We express the cohomology of the tensor product $_n{\cal K}_{E_1}\otimes {_n{\cal K}_{E_2}}$ in terms of cohomology of the symmetric powers of $X$. This may be considered as a geometric interpretation of the local part of the classical Rankin-Selberg method for GL(n) in the framework of the geometric Langlands program.

math.AG↗