Affine Matsuki correspondence for sheaves

TL;DR

This paper extends the affine Matsuki correspondence to an equivalence of derived categories of sheaves, utilizing Morse theory and fusion structures to relate real and symmetric loop group orbits in affine Grassmannians.

Contribution

It introduces a novel categorical equivalence lifting the affine Matsuki correspondence, incorporating fusion structures and Morse theory in the affine setting.

Findings

Established an equivalence of derived categories of sheaves

Connected real and symmetric loop group orbits via categorical methods

Linked affine Grassmannian sheaves with Schubert constructible sheaves

Abstract

We lift the affine Matsuki correspondence between real and symmetric loop group orbits in affine Grassmannians to an equivalence of derived categories of sheaves. In analogy with the finite-dimensional setting, our arguments depend upon the Morse theory of energy functions obtained from symmetrizations of coadjoint orbits. The additional fusion structures of the affine setting lead to further equivalences with Schubert constructible derived categories of sheaves on real affine Grassmannians.