On total Springer representations

Abstract

This thesis studies the alternating sum of cohomology groups of a Springer fiber (in characteristic 0), called a total Springer representation, as a representation of both the Weyl group and the stabilizer of the corresponding nilpotent element. For classical types, we present explicit formulas for the decomposition of total Springer representations into irreducible ones of the corresponding Weyl group using Kostka-Foulkes polynomials. Also, the character value at any element contained in the maximal parabolic subgroup(s) of type A is explicitly given in terms of Green polynomials. As a result, closed formulas for the Euler characteristic of Springer fibers are deduced. Our proof relies on analysis of geometry of Springer fibers and combinatorics of symmetric functions. Moreover, we provide formulas for the character value of a total Springer representation at any element in the stabilizer of the corresponding nilpotent element. For exceptional types, the character values of total Springer representations are completely known. Here, we only describe the decomposition of such representations into irreducible ones of stabilizers of corresponding nilpotent elements.