# Hecke operators on quasimaps into horospherical varieties

### Dennis Gaitsgory

Department of Mathematics Department of Mathematics Harvard University Northwestern University Cambridge Evanston MA 02138 IL 60208### David Nadler

Department of Mathematics Department of Mathematics Harvard University Northwestern University Cambridge Evanston MA 02138 IL 60208

## Abstract

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 $X_{horo}$, the category $Q(Z_{horo})$ is equivalent to a category of vector spaces graded by a lattice. Thus the associated subgroup $Hˇ_{horo}$ 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 $X_{horo}$. Its associated subgroup $Hˇ_{horo}$ turns out to be a maximal torus in the subgroup $Hˇ$ associated to $X$.

## Cite this article

Dennis Gaitsgory, David Nadler, Hecke operators on quasimaps into horospherical varieties. Doc. Math. 14 (2009), pp. 19–46

DOI 10.4171/DM/264