  The aim of this paper is to identify a certain tensor category of perverse
sheaves on the real loop Grassmannian of a real form $G_{\mathbb R}$ of a
connected reductive complex algebraic group $G$ with the category of
finite-dimensional representations of a connected reductive complex algebraic
subgroup $\check H$ of the dual group $\check G$. The root system of $\check H$
is closely related to the restricted root system of the real form $G_{\mathbb
R}$. The fact that $\check H$ is reductive implies that an interesting family
of real algebraic maps satisfies the conclusion of the Decomposition Theorem of
Beilinson-Bernstein-Deligne.
