DOCUMENTA MATHEMATICA, Vol. 14 (2009), 19-46

Dennis Gaitsgory and David Nadler

Hecke Operators on Quasimaps into Horospherical Varieties

Let $G$ be a connected reductive complex algebraic group. This paper and its companion \cite{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 $X$.

2000 Mathematics Subject Classification: 22E67, 14M17

Keywords and Phrases: Loop spaces, spherical varieties, Langlands duality

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