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 (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.