Weyl group characters afforded by zero weight spaces.

Mark Reeder (Boston College)

Let G denote a compact Lie group and $T$ a maximal torus in $G$ .

If $V$ is an irreducible representation of $G$ then the zero weight-space $V^T$ affords a representation of the Weyl group $W$ of $T$ . The problem of describing the mapping $V \longrightarrow V^T$ from $G$ -representations to $W$ -representations attracted the interest of Kostant and (independently) Gutkin, in the 1970’s.

Kostant proved two fundamental results: (i) He gave a formula for $dim V^T$ and (ii) he proved that the trace of a Coxeter element on $V^T$ is either $0$ , $1$ or $-1$ , and he gave a formula for the exact value.

First I will explain why the $W$ -action on $V^T$ is of interest for harmonic analysis on $G$ , and then give some new results:

1. A formula for the trace on $V^T$ of any element of $W$ , generalizing Kostant’s result (i) for the identity element of $W$ .

2. A generalization of Kostant’s result (ii) for all elliptic regular elements of $W$ . This relies on a characterization of elliptic regular elements in $W$ , using Thomae’s function on a Lie group. (I promise to explain all of this.)

3. If the long element in $W$ equals $-1$ then it is elliptic regular. In these cases, 2. gives a method for determining when $V^T$ is irreducible for $W$ . This addresses a question asked by Humphreys, and is related to an earlier question posed by Kostant to Lusztig, when they first met.