  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.
