Let be a connected reductive complex algebraic group. This paper and its companion citeGNcombo06 are devoted to the space of meromorphic quasimaps from a curve into an affine spherical -variety . The space may be thought of as an algebraic model for the loop space of . The theory we develop associates to a connected reductive complex algebraic subgroup of the dual group . The construction of is via Tannakian formalism: we identify a certain tensor category of perverse sheaves on with the category of finite-dimensional representations of . In this paper, we focus on horospherical varieties, a class of varieties closely related to flag varieties. For an affine horospherical -variety , the category is equivalent to a category of vector spaces graded by a lattice. Thus the associated subgroup 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 -variety , one may associate a horospherical variety . Its associated subgroup turns out to be a maximal torus in the subgroup associated to .
Cite this article
Dennis Gaitsgory, David Nadler, Hecke operators on quasimaps into horospherical varieties. Doc. Math. 14 (2009), pp. 19–46DOI 10.4171/DM/264