A Survey on Springer Theory

Abstract

This is not standard in the sense that we understand a Springer map to be a collapsing of homogeneous bundles. Apart from that we use mostly techniques from Chriss and Ginzbergs book but we work in the equivariant derived category of Bernstein and Lunts. We define Steinberg algebras as (equivariant) Borel-Moore homology algebras of the associated Steinberg varieties. The data of the BBD-decomposition theorem applied to the Springer map give a parametrization of projective graded and simple graded modules over the Steinberg algebra. Also, the projective graded modules are equivalent to a category of shifts of perverse sheaves. This has as a consequence for example the Springer correspondence. We call classical Springer Theory what is usually considered as Springer Theory. The main results are parametrizations of simple modules of different types of Hecke algebras. Our second main example is quiver-graded Springer theory (due to Lusztig), here the Steinberg algebras are the quiver Hecke algebras. We also explain Lusztig's and Khovanov-Lauda's monoidal categorification of the negative half of the quantum group using the categories of shifts of perverse sheaves and projective graded modules over the quiver Hecke algebra respectively.