Exterior powers of the reflection representation in the cohomology of Springer fibres

TL;DR

This paper determines the degrees in which exterior powers of the reflection representation appear in the cohomology of Springer fibers for certain nilpotent elements, partially confirming a conjecture and extending previous results.

Contribution

It identifies the degrees of exterior powers in Springer fiber cohomology under specific conditions, extending known results and verifying a conjecture.

Findings

Determines degrees of exterior powers in cohomology for regular nilpotent elements in Levi subalgebras.

Shows that a certain invariant algebra is a free exterior algebra.

Partially verifies Lehrer--Shoji conjecture for types A, B, C.

Abstract

Let $H^*(\calB_e)$ be the cohomology of the Springer fibre for the nilpotent element $e$ in a simple Lie algebra $\g$, on which the Weyl group $W$ acts by the Springer representation. Let $\Lambda^i V$ denote the $i$th exterior power of the reflection representation of $W$. We determine the degrees in which $\Lambda^i V$ occurs in the graded representation $H^*(\calB_e)$, under the assumption that $e$ is regular in a Levi subalgebra and satisfies a certain extra condition which holds automatically if $\g$ is of type A, B, or C. This partially verifies a conjecture of Lehrer--Shoji, and extends the results of Solomon in the $e=0$ case and Lehrer--Shoji in the $i=1$ case. The proof proceeds by showing that $(H^*(\calB_e) \otimes \Lambda^* V)^W$ is a free exterior algebra on its subspace $(H^*(\calB_e)\otimes V)^W$.