Exterior powers of the reflection representation in Springer theory

Abstract

We give a proof of a conjecture of Lehrer and Shoji regarding the occurrences of the exterior powers of the reflection representation in the cohomology of Springer fibers. The actual theorem proved is a slight extension of the original conjecture to all nilpotent orbits and also takes into account the action of the component group. The method is to use Shoji's approach to the orthogonality formulas for Green functions to relate the symmetric algebra to a sum over Green functions. In the second part of the paper we give an explanation of the appearance of the Orlik-Solomon exponents using a result from rational Cherednik algebras.