Remarks on Springer's representations

TL;DR

This paper characterizes certain irreducible Weyl group representations linked to nilpotent orbits in reductive Lie algebras and addresses Serre's question on unipotent element conjugacy classes.

Contribution

It provides an a priori description of Weyl group representations parametrizing nilpotent orbits and resolves Serre's question on conjugacy classes of unipotent elements.

Findings

Explicit description of Weyl group representations for nilpotent orbits

Answer to Serre's question on unipotent element conjugacy classes

Applicable in arbitrary characteristic

Abstract

We give an apriori description of a set of irreducible representations of a Weyl group which parametrize the nilpotent orbits in the Lie algebra of a connected reductive group in arbitrary characteristic. We also answer a question of Serre concerning the conjugacy class of a power of a unipotent element in a connected reductive group.