  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 an algebraic model for the loop
space of X. In this paper, we associate 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.
