# Hecke Operators on Quasimaps into Horospherical Varieties

 Title: Hecke Operators on Quasimaps into Horospherical Varieties Author: Gaitsgory, Dennis; Nadler, David Note: Order does not necessarily reflect citation order of authors. Citation: Gaitsgory, Dennis, and David Nadler. 2009. Hecke operators on quasimaps into horospherical varieties. Documenta Mathemathica 14: 19-46. Full Text & Related Files: Gaitsgory_HeckeOperators.pdf (281.4Kb; PDF) 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. Published Version: http://www.emis.ams.org/journals/DMJDMV/vol-14/02.pdf Other Sources: http://arxiv.org/abs/math/0411266 Terms of Use: This article is made available under the terms and conditions applicable to Open Access Policy Articles, as set forth at http://nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of-use#OAP Citable link to this page: http://nrs.harvard.edu/urn-3:HUL.InstRepos:10039806 Downloads of this work: