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

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

2010-01-18

arxiv.org/abs/1001.3164v1

C. R. Math. Acad. Sci. Paris 348 (2010), no. 19-20, 1055-1058

Mathematics

Representation Theory

8 pages

Scientific paper

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 $(H^*(\calB_e)\otimes V)^W$.

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