Hecke Operators on Quasimaps into Horospherical Varieties

Abstract

Let G be a connected reductive complex algebraic group. This paper and its companion [GN06] 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_{horo}), the category Q((Z_{horo})) is equivalent to a category of vector spaces graded by a lattice. Thus the associated subgroup (\check H_{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 (\check H_{horo}) turns out to be a maximal torus in the subgroup (\check H) associated to X.