On the reflection representation in Springer's theory

Abstract

The theory of arrangements of hyperplanes allows to attach to every parabolic subgroup of a finite Coxeter group some numbers which look very much like exponents of the Coxeter group. In the case of Weyl groups similar numbers arise from the character theory of finite groups of Lie type, and more generally from the theory of Springer's representations. For exceptional Weyl groups these numbers were known to coincide, at least if the characteristic is not a small prime. Lehrer and Shoji [8] have shown that in characteristic 0 the same is true for classical Weyl groups, by computing the multiplicity of the reflection representation in the Springer representations associated to various nilpotent orbits. According to a note added in proof, they can handle all nilpotent orbits which are relevant to the connection with arrangements of hyperplanes, but some nilpotent orbits still evade their investigation.