Hecke operators on quasimaps into horospherical varieties

Let $G$ be a connected reductive complex algebraic group. This paper is part of a project 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 $\check H$. Combinatorial shadows of the group $\check H$ govern many aspects of the geometry of $X$ such as its compactifications and invariant differential operators. When $X$ is a symmetric variety, the group $\check H$ coincides with that associated to the corresponding real form of $G$ via the (real) geometric Satake correspondence. In this paper, we focus on horospherical varieties, a class of varieties closely related to flag varieties.