Stabilisers of eigenvectors of finite reflection groups

TL;DR

This paper investigates the stabilizers of eigenvectors in finite reflection groups, providing bounds on their root systems and connecting these results to Lie theory and semisimple conjugacy classes.

Contribution

It generalizes Kostant's result by bounding the roots in stabilizer subgroups and offers a Lie-theoretic interpretation relevant to conjugacy classes.

Findings

Upper bound for roots in stabilizer subgroup of eigenvectors

Generalization of Kostant's regular eigenvector result

Lie-theoretic interpretation related to semisimple conjugacy classes

Abstract

Let $x$ be an eigenvector for an element of a finite irreducible reflection group $W$. Let $W_x$ denote the subgroup of $W$ which stabilises $x$. We provide an upper bound for the number of roots in the root system of $W_x$ . This generalises a result of Kostant, who showed that every eigenvector with eigenvalue a primitive $h^\mathrm{th}$ root of unity is regular, where $h$ is the Coxeter number of $W$. We also give a Lie-theoretic interpretation of our result in the study of semisimple conjugacy classes over Laurent series. In a forthcoming paper, we use this result to establish a geometric analogue of a conjecture of Gross and Reeder.