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

Abstract

Let H ⁎ ( B e ) be the cohomology of the Springer fibre for the nilpotent element e in a simple Lie algebra g . Let Λ i V denote the ith exterior power of the reflection representation of W. We determine the degrees in which Λ i V occurs in the graded representation H ⁎ ( B 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 and Shoji.