Involutions on the affine Grassmannian and moduli spaces of principal bundles

Let G be a simply connected semisimple group over 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 SL(2) on the framed moduli space of Gmequivariant principal G-bundles on P. As a result, the fixed-point set of the involution can be partitioned into strata indexed by conjugacy classes of homomorphisms N → G where N is the normalizer of Gm in SL(2). When G = SL(r), the strata are Nakajima quiver varieties M 0 (v,w) of type D.