A generalization of combinatorial identities for stable discrete series constants

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 discrete series character identities proved in a previous paper of the second author. 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 Appendix A of the Goresky--Kottwitz--MacPherson article. In Appendix B we extend the notion of $2$-structures (due to Herb) to pseudo-root systems.