Involutions on the affine Grassmannian and moduli spaces of principal bundles

Friday 20 Nov, 12-1pm, Place: Carslaw 375

Let $G$ be a connected reductive group over $\mathbb{C}$. For reasons related to our work on nilpotent orbits, Pramod Achar and I were led to study a certain involution of an open subset of the affine Grassmannian of $G$. In this talk I will explain a new interpretation of this involution: it 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 $\mathrm{SL}(2)$. In the case where $G=\mathrm{GL}(r)$, these strata are isomorphic to quiver varieties of type D.