Weyl group characters afforded by zero weight spaces

Mark Reeder

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arXiv · 1910.02463

Weyl group characters afforded by zero weight spaces

Mark Reeder

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Abstract

Let $G$ be a simple complex Lie group with Weyl group $W$. We give a formula for the character of $W$ on the zero weight space of any finite dimensional representation of $G$. The formula involves partition functions, generalizing Kostant's partition function. On the elliptic set of $W$ the partition functions are trivial. On the elliptic regular set, the character formula is a monomial product of certain co-roots, up to a constant equal to $0$ or $\pm 1$. For a Coxeter element we recover Kostant's formula for this trace. If the long element $w_0=-1$, our formula leads to a method for determining all representations of $G$ for which the zero weight space is irreducible.

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BibTeX RIS

Mark Reeder. 2019-10-06. Weyl group characters afforded by zero weight spaces. https://arxiv.org/abs/1910.02463

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