A geometric approach to characters of Hecke algebras

TL;DR

This paper introduces a geometric method to analyze characters of Hecke algebras using Lusztig varieties and intersection cohomology, providing new insights and generalizations in representation theory.

Contribution

It develops a geometric approach to character formulas of Hecke algebras, generalizes the Brosnan-Chow solution, and connects algebraic and combinatorial structures.

Findings

Frobenius character matches symmetric functions for type A

Cellular decompositions help determine local invariant cycle maps

Generalizes Lusztig's results on character sheaves

Abstract

To any element of a connected, simply connected, semisimple complex algebraic group G and a choice of an element of the corresponding Weyl group there is an associated Lusztig variety. When the element of G is regular semisimple, the corresponding variety carries an action of the Weyl group on its (equivariant) intersection cohomology. From this action, we recover the induced characters of an element of the Kazhdan-Lusztig basis of the corresponding Hecke algebra. In type A, we prove a more precise statement: that the Frobenius character of this action is precisely the symmetric function given by the characters of a Kazhdan-Lusztig basis element. The main idea is to find celular decompositions of desingularizations of these varieties and apply the Brosnan-Chow palindromicity criterion for determining when the local invariant cycle map is an isomorphism. This recovers some results of Lusztig about character sheaves and gives a generalization of the Brosnan-Chow solution to the Sharesian-Wachs conjecture to non-codominant permutations, where singularities are involved. We also review the connections between Immanants, Hecke algebras, and Chromatic quasisymmetric functions of indifference graphs.