slightly different character identity for averaged discrete series characters of $G({{\mathbb{R}}})$, also involving discrete series characters of the endoscopic groups of $G$, and whose relationship with the Goresky–Kottwitz–MacPherson identity was not clear.

We return to a discussion of the current article.

In a previous article [@Ehrenborg_Morel_Readdy], we investigate the character identity of [@Morel]. In particular we relate it to the geometry of the Coxeter complex of the symmetric group and give a simpler and more natural proof than the brute force calculation in the appendix of [@Morel]. The goal of the present article is to generalize the approach and methods of [@Ehrenborg_Morel_Readdy] and to prove a combinatorial identity (Theorem \[thm\_main\]) that implies the character formulas of [@GKM] and of [@Morel] (see Section \[section\_applications\]). To obtain the character formula of [@GKM] from our results, we need to use Herb’s formula for averaged discrete series characters (see for example [@Herb] and [@Herb-2S]). We also generalize, in Corollary \[cor\_shelling\_order\] and Lemma \[lemma\_Coxeter\_A\], the geometric result of [@Ehrenborg_Morel_Readdy] (see Theorem 4.3 of that article). In fact, we prove an identity for all Coxeter systems with finite Coxeter groups, and not just the root systems that are generated by strongly orthogonal roots. The representation-theoretic interpretation 