Hecke operators on quasimaps into horospherical varieties

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 $\check{H}$ of the dual group $\check{G}$. The construction of $\check{H}$ 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 $\check{H}$. In this paper, we focus on horospherical varieties, a class of varieties closely related to flag varieties. For an affine horospherical $G$-variety $X_{\mathrm{horo}}$, the category $Q(Z_{\mathrm{horo}})$ is equivalent to a category of vector spaces graded by a lattice. Thus the associated subgroup ${\check{H}}_{\mathrm{horo}}$ 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 $X_{\mathrm{horo}}$. Its associated subgroup ${\check{H}}_{\mathrm{horo}}$ turns out to be a maximal torus in the subgroup $\check{H}$ associated to $X$.