This article is concerned with the constants that appear in Harish-Chandra's character formula for stable discrete series of real reductive groups, although it does not require any knowledge about real reductive groups or discrete series. In Harish-Chandra's work the only information we have about these constants is that they are uniquely determined by an inductive property. Later, Goresky-Kottwitz-MacPherson (1997) and Herb (2000) gave different formulas for these constants. In this article, we generalize these formulas to the case of arbitrary finite Coxeter groups (in this setting, discrete series no longer make sense), and give a direct proof that the two formulas agree. We actually prove a slightly more general identity that also implies the combinatorial identity underlying the discrete series character identities of Morel (2011). We deduce this identity from a general abstract theorem giving a way to calculate the alternating sum of the values of a valuation on the chambers of a Coxeter arrangement. We also introduce a ring structure on the set of valuations on polyhedral cones in Euclidean space with values in a fixed ring. This gives a theoretical framework for the valuation appearing in Goresky- Kottwitz-MacPherson's 1997 paper. In an appendix, we extend Herb's notion of 2-structures to pseudo-root systems.
Bibliographical noteFunding Information:
Acknowledgments. We thank the anonymous referees for their helpful comments. The greatest part of this work was written while the second author was a professor at Princeton University. It was also partially supported by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program “Investissements d’Avenir” (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR). More precisely, the authors would like to thank the École Normale Supérieure de Lyon (ÉNS de Lyon) for its hospitality and support to the second author during the academic year 2017–2018, and to the first and third author during one week visits.
Funding. This work was partially supported by grants from the Simons Foundation (#429370 to Richard Ehrenborg and #422467 to Margaret Readdy).
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- Averaged discrete series characters
- Coxeter systems
- hyperplane arrangements
ASJC Scopus subject areas
- Algebra and Number Theory
- Discrete Mathematics and Combinatorics