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 $\mathrm{S}\mathrm{L}(2)$ 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 $\mathrm{S}\mathrm{L}(2)$. When $G=\mathrm{S}\mathrm{L}(r)$, the strata are Nakajima quiver varieties ${\mathfrak{M}}_0^{\mathrm{r}\mathrm{e}\mathrm{g}}(\mathbf{v},\mathbf{w})$ of type D.