Perverse Sheaves on Real Loop Grassmannians

Abstract

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.

63 pages; final version, to appear in Invent math