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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; see [@GKM Theorem 3.1] and [@Herb-2S Theorem 4.2]. 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 Proposition 3.3.1]. We also introduce a signed convolution of valuations on polyhedral cones in Euclidean space and show that the resulting function is a valuation. This gives a theoretical framework for the valuation appearing in [@GKM Appendix A]. In Appendix \[appendix\_2\_structures\] we extend the notion of $2$-structures (due to Herb) to pseudo-root systems.'
author:
- '[Richard EHRENBORG]{}, [Sophie MOREL]{} and [Margaret READDY]{}'
bibliography:
- 'TS2\_short.bib'
date: '.'
title: 