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Let $G$ be a simply connected semisimple group over $\mathbb{C}$. We showthat 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 thenontrivial 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 aresult, the fixed-point set of the involution can be partitioned into strataindexed by conjugacy classes of homomorphisms $N\to G$ where $N$ is thenormalizer of $\mathbb{G}_m$ in $\mathrm{SL}(2)$. When $G=\mathrm{SL}(r)$, thestrata are Nakajima quiver varieties$\mathfrak{M}_0^{\mathrm{reg}}(\mathbf{v},\mathbf{w})$ of type D.

Involutions on the affine Grassmannian and moduli spaces of principal bundles