Perverse sheaves on symmetric products of the plane, Schur algebras, and K-theory

In analogy with the (generalized) Springer correspondence relating perverse sheaves on a nilpotent cone to representations of the Weyl group, we consider perverse sheaves on the symmetric product of n copies of the plane C², constructible with respect to the natural stratification by collision of points. This category is semisimple when the coefficients have characteristic zero, but with positive characteristic coefficients it can be very complicated. We show that this category is equivalent to modules over a new finite-dimensional algebra R_n, given by K-theory of equivariant sheaves on the symmetric group S_n, under the action of Young subgroups on the left and right. I will explain how this algebra arises using the K-theory of Hilbert schemes and a theorem of Bridgeland, King, and Reid. I will also explain a relation between R_n and Schur algebras, which play an important role in the modular representation theory of the symmetric group. Joint work with Carl Mautner.