Matsuki duality for loop groups

TL;DR

This paper extends Matsuki duality to loop groups, establishing a bijection between symmetric and real polynomial loop group orbits on affine Grassmannians and flag varieties, with applications to orbit parametrizations and vector bundles.

Contribution

It introduces a version of Matsuki duality for loop groups, linking symmetric and real polynomial loop group orbits on affine geometric structures.

Findings

Established bijection between symmetric and real polynomial loop group orbits

Provided orbit parametrizations for affine Grassmannians and flag varieties

Connected orbit structures with vector bundles and Kottwitz sets

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 .