Dennis Gaitsgory and David Nadler

摘要

Let $G$ be a connected reductive complex algebraic group. This paper and its companioncite{GNcombo06} 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 $catq(Z)$ 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 $X_{on{horo}}$, the category $catq(Z_{on{horo}})$ is equivalent to a category of vector spaces graded by a lattice. Thus the associated subgroup $check H_{on{horo}}$ 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 $X_{on{horo}}$. Its associated subgroup $check H_{on{horo}}$ turns out to be a maximal torus in the subgroup $check H$ associated to $X$.