recursion result for arbitrary hyperplane arrangements. The main technical tool is Theorem \[theorem\_signed\_convolution\], which generalizes [@GKM Proposition A.4].

Section \[section\_main\_thm\] contains the statement and proof of our main theorem (Theorem \[thm\_main\]). We specialize to the case of Coxeter systems and introduce the second half of our main identity, generalizing a construction of Herb that we review in detail in Appendix \[appendix\_2\_structures\]. The proof that both halves of the identity are equal follows the same lines as the proofs of the character formulas in the Goresky–Kottwitz–MacPherson and Herb articles: First, using the recursive expressions for both sides (Propositions \[prop\_induction\_formula\_for\_psi\] and \[prop\_induction\_RHS\]), we reduce to the case where $\lambda$ is dominant so that $H_\lambda$ is as far as possible from the base chamber. Then we treat this case directly by a geometric argument based on the results of Section \[section\_hyperplane\_arrangements\].

In Section \[section\_applications\] we explain how Theorem \[thm\_main\] implies the identities of [@GKM Theorem 3.1] and of [@Ehrenborg_Morel_Readdy Theorem 6.4], and in Section \[section\_concluding\_remarks\] we include concluding remarks.

We finish with two appendices. Each can be read independently from the rest of the article. The goal of our Appendix \[appendix\_valuations\] is to generalize [@GKM Proposition A.4], which is a key part of 