Exterior powers of the reflection representation in Springer theory

TL;DR

This paper proves a conjecture about how exterior powers of the reflection representation appear in the cohomology of Springer fibers, extending it to all nilpotent orbits and considering component group actions.

Contribution

It extends Lehrer and Shoji's conjecture to all nilpotent orbits and incorporates the action of the component group, using Shoji's orthogonality formulas and rational Cherednik algebra insights.

Findings

Confirmed the conjecture for all nilpotent orbits

Connected Green functions to symmetric algebra representations

Explained Orlik-Solomon exponents via Cherednik algebras

Abstract

We give a proof of a conjecture of Lehrer and Shoji regarding the occurrences of the exterior powers of the reflection representation in the cohomology of Springer fibers. The actual theorem proved is a slight extension of the original conjecture to all nilpotent orbits and also takes into account the action of the component group. The method is to use Shoji's approach to the orthogonality formulas for Green functions to relate the symmetric algebra to a sum over Green functions. In the second part of the paper we give an explanation of the appearance of the Orlik-Solomon exponents using a result from rational Cherednik algebras.