Spherical varieties and Langlands duality

Let $G$ be a connected reductive complex algebraic group. This paper is 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 a finite-dimensional 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 $\mathbf Q(Z)$ of perverse sheaves on $Z$ with the category of finite-dimensional representations of $\check H$. The group $\check H$ encodes many aspects of the geometry of $X$.