Combinatorics of hyperplane arrangements and Witten zeta function at the origin

We introduce a new method that brings the combinatorics of hyperplane arrangements into the study of representation zeta functions of compact Lie groups. For the Witten zeta function $ζ_Φ(s)$ associated with a root system $Φ$, our method yields elegant formulas for $ζ_Φ(0)$ and $ζ_Φ'(0)$ in terms of the exponents of various parabolic subsystems of $Φ$. Such formulas do not appear to be readily accessible through the conventional analytic techniques in the literature. More generally, the method applies to a broad family of conical zeta functions, expressing these two special values through the Möbius function of the intersection poset of the associated hyperplane arrangement.