of our identity in the general case is still unclear.

We now describe in more detail the different sections of the article.

In Section \[section\_hyperplane\_arrangements\] we review some background material about real hyperplane arrangements and introduce our main geometric construction, which we call the *weighted complex*. The weighted complex is the set of all the faces of a fixed hyperplane arrangement that are on the nonnegative side of an auxiliary hyperplane $H_\lambda$. It contains what is known as the *bounded complex* in the theory of affine oriented matroids, and coincides with it if $H_\lambda$ is in general position. We also prove that, under a hypothesis about the dihedral angles between the hyperplanes of the arrangement (Condition \[(A)\] in Subsection \[section\_weighted\_complex\], which is always true in the Coxeter case), the weighted complex is shellable.

In Section \[section\_definition\_sum\] we introduce one half of our main identity (see Definition \[def\_psi\]) and study its properties. In particular, we establish a recursion result in Proposition \[prop\_induction\_formula\_for\_psi\] which will be the heart of the proof of Theorem \[thm\_main\]. Our treatment is inspired by that of Sections 1-2 and Appendix A of [@GKM], but we make no assumptions on the Coxeter system. In fact, we prove the 