Involutions on the affine Grassmannian and moduli spaces of principal bundles

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.