Matsuki duality for loop groups

Abstract

We establish versions of Matsuki duality for loop groups. The main result is a bijection between symmetric loop group orbits and real polynomial loop group orbits on the affine Grassmannians or affine flag varieties. Along the way we obtain orbit parametrizations and make connections with vector bundles on real and twistor-${\mathbb{P}}^1$ and Kottwitz sets .

31 pages