Involutions on the affine Grassmannian and moduli spaces of principal bundles
speaker: Anthony Henderson (University of Sydney)abstract: Let $G$ be a semisimple algebraic group over $\mathbb{C}$. In previous work, Pramod Achar and I discovered a relationship between some of the nilpotent orbits of $G$ and some of the Schubert cells in the affine Grassmannian $\mathsf{Gr}$ of $G$, generalizing well-known results for $\mathrm{SL}(n)$. This relationship involved a certain involution on an open subset of $\mathsf{Gr}$. 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 consequence, 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 $G=\mathrm{SL}(n)$, these strata are isomorphic to quiver varieties of type D.
timetable:
Tue 14 Jun, 16:15 - 17:30, Aula Bianchi Scienze
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