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.