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Let $H^*(\calB_e)$ be the cohomology of the Springer fibre for the nilpotentelement $e$ in a simple Lie algebra $\g$, on which the Weyl group $W$ acts bythe Springer representation. Let $\Lambda^i V$ denote the $i$th exterior powerof the reflection representation of $W$. We determine the degrees in which$\Lambda^i V$ occurs in the graded representation $H^*(\calB_e)$, under theassumption that $e$ is regular in a Levi subalgebra and satisfies a certainextra condition which holds automatically if $\g$ is of type A, B, or C. Thispartially verifies a conjecture of Lehrer--Shoji, and extends the results ofSolomon in the $e=0$ case and Lehrer--Shoji in the $i=1$ case. The proofproceeds by showing that $(H^*(\calB_e) \otimes \Lambda^* V)^W$ is a freeexterior algebra on its subspace $(H^*(\calB_e)\otimes V)^W$.

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