Hecke operators on quasimaps into horospherical varieties

Dennis Gaitsgory ; David Nadler

Documenta mathematica, Tome 14 (2009), pp. 19-46

Résumé

Let G be a connected reductive complex algebraic group. This paper and its companion citeGNcombo06 are devoted to the space Z of meromorphic quasimaps from a curve into an affine spherical G-variety X. The space Z may be thought of as an algebraic model for the loop space of X. The theory we develop associates to X a connected reductive complex algebraic subgroup Hˇ of the dual group Gˇ. The construction of Hˇ is via Tannakian formalism: we identify a certain tensor category Q(Z) of perverse sheaves on Z with the category of finite-dimensional representations of Hˇ. In this paper, we focus on horospherical varieties, a class of varieties closely related to flag varieties. For an affine horospherical G-variety Xhoro​, the category Q(Zhoro​) is equivalent to a category of vector spaces graded by a lattice. Thus the associated subgroup Hˇhoro​ is a torus. The case of horospherical varieties may be thought of as a simple example, but it also plays a central role in the general theory. To an arbitrary affine spherical G-variety X, one may associate a horospherical variety Xhoro​. Its associated subgroup Hˇhoro​ turns out to be a maximal torus in the subgroup Hˇ associated to X.

Détail

DOI : 10.4171/dm/264

Classification : 14M17, 22E67
Mots-clés : spherical varieties, loop spaces, Langlands duality