Involutions on the affine Grassmannian and moduli spaces of principal bundles

TL;DR

This paper links an involution on the affine Grassmannian of a semisimple group to Weyl group actions on moduli spaces of principal bundles, revealing stratifications related to quiver varieties of type D.

Contribution

It establishes a geometric correspondence between an involution on the affine Grassmannian and Weyl group actions on moduli spaces, identifying fixed points with Nakajima quiver varieties.

Findings

Involution corresponds to Weyl group action on moduli space

Fixed points stratified by conjugacy classes of homomorphisms

Strata are Nakajima quiver varieties of type D

Abstract

Let $G$ be a simply connected semisimple group over $\mathbb{C}$. We show that a certain involution of an open subset of the affine Grassmannian of $G$, defined previously by Achar and the author, corresponds to the action of the nontrivial Weyl group element of $\mathrm{SL}(2)$ on the framed moduli space of $\mathbb{G}_m$-equivariant principal $G$-bundles on $\mathbb{P}^2$. As a result, the fixed-point set of the involution can be partitioned into strata indexed by conjugacy classes of homomorphisms $N\to G$ where $N$ is the normalizer of $\mathbb{G}_m$ in $\mathrm{SL}(2)$. When $G=\mathrm{SL}(r)$, the strata are Nakajima quiver varieties $\mathfrak{M}_0^{\mathrm{reg}}(\mathbf{v},\mathbf{w})$ of type D.