University of Sydney Algebra Seminar

Anthony Henderson (University of Sydney)

Friday 9 September, 12:05-12:55pm, Carslaw 175

The affine Grassmannian and the nilpotent cone

Let \(G\) be a simply-connected simple algebraic group over \(\mathbb{C}\). The geometric Satake correspondence is a category equivalence between representations of the dual group \(G^\vee\) and \(G(\mathbb{C}[[t]])\)-equivariant perverse sheaves on the affine Grassmannian of \(G\). The Springer correspondence is an equivalence between representations of the Weyl group \(W\) and a subcategory of \(G\)-equivariant perverse sheaves on the nilpotent cone of \(G\). The obvious functor from representations of \(G^\vee\) to representations of \(W\), namely taking invariants for the maximal torus, seems difficult to describe in geometric terms. However, I will explain a simple description of the restriction of this functor to the category of "small" representations of \(G^\vee\), in terms of a new relationship between the "small part" of the affine Grassmannian and the nilpotent cone. This is joint work with Pramod Achar (Louisiana State University).