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.