Abstract
Let G be a compact 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 weighted 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 coroots, up to a constant equal to 0 or ± 1. This generalizes Kostant’s formula for the trace of a Coxeter element on a zero weight space. If the long element w0 = − 1, our formula gives a method for determining all representations of G for which the zero weight space is irreducible.
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Notes
The terminology “regular in W” follows Springer [35]. It differs from the notion of regularity of elements of G: If w is regular in W then the elements of wG need not be regular in G.
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In memory of Bert Kostant
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Reeder, M. Weyl Group Characters Afforded By Zero Weight Spaces. Transformation Groups (2022). https://doi.org/10.1007/s00031-022-09709-9
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DOI: https://doi.org/10.1007/s00031-022-09709-9