A generalization of combinatorial identities for stable discrete series constants

TL;DR

This paper generalizes formulas for constants in Harish-Chandra's character formula from real reductive groups to arbitrary finite Coxeter groups, providing a unified combinatorial framework and proving their equivalence.

Contribution

It extends known formulas for stable discrete series constants to Coxeter groups and introduces a valuation-based framework for their calculation.

Findings

Proves the equivalence of different formulas for Coxeter group constants.

Introduces a valuation-based approach to compute chamber sums.

Extends the concept of 2-structures to pseudo-root systems.

Abstract

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 and Herb 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. 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 Appendix A of the Goresky-Kottwitz-MacPherson paper. In Appendix B we extend the notion of $2$-structures (due to Herb) to pseudo-root systems.