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    A generalization of combinatorial identities\
    for stable discrete series constants
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Introduction
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Although this paper deals exclusively with the combinatorics of real hyperplane arrangements and Coxeter complexes, it has its origin in the representation theory of real reductive groups and its connections with the cohomology of locally symmetric spaces, and in particular of Shimura varieties. We start by explaining some of this background. This explanation can be safely skipped by the reader not interested in Shimura varieties.

Let $G$ be an algebraic group over ${{\mathbb{Q}}}$. To simplify the exposition, we assume that $G$ is connected and semisimple. Let $K_\infty$ be a maximal compact subgroup of $G({{\mathbb{R}}})$ and $K$ be an open compact subgroup of $G(\mathbb{A}^\infty)$, where $\mathbb{A}^\infty=\widehat{{{\mathbb{Z}}}}\otimes_{{\mathbb{Z}}}{{\mathbb{Q}}}$ is the ring of finite adèles of ${{\mathbb{Q}}}$. We consider the double quotient $X_K=G({{\mathbb{Q}}})\backslash(G({{\mathbb{R}}})\times
G(\mathbb{A}^\infty))/(K_\infty\times K)$. This is a real analytic variety for $K$ small enough, and the projective system $(X_K)_{K\subset
G(\mathbb{A}^\infty)}$ has an action of $G(\mathbb{A}^\infty)$ by Hecke correspondences, which induces an action of the Hecke algebra at level $K$ on the cohomology of $X_K$, for any reasonable cohomology theory.

We restrict our attention further to the case where the real Lie group $G({{\mathbb{R}}})$ has a discrete 