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

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 $Λ^i V$ denote the $i$th exterior power of the reflection representation of $W$. We determine the degrees in which $Λ^i V$ 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 $(H^*(\calB_e) \otimes Λ^* V)^W$ is a free exterior algebra on its subspace $(H^*(\calB_e)\otimes V)^W$.