  In the computation of the intersection cohomology of Shimura varieties, or of
the $L^2$ cohomology of equal rank locally symmetric spaces, combinatorial
identities involving averaged discrete series characters of real reductive
groups play a large technical role. These identities can become very
complicated and are not always well-understood (see for example the appendix of
[8]). We propose a geometric approach to these identities in the case of Siegel
modular varieties using the combinatorial properties of the Coxeter complex of
the symmetric group. Apart from some introductory remarks about the origin of
the identities, our paper is entirely combinatorial and does not require any
knowledge of Shimura varieties or of representation theory.
