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 ${\zeta }_{\mathrm{\Phi }}(s)$ associated with a root system $\mathrm{\Phi }$, our method yields elegant formulas for ${\zeta }_{\mathrm{\Phi }}(0)$ and ${\zeta }_{\mathrm{\Phi }}^{\mathrm{\prime }}(0)$ in terms of the exponents of various parabolic subsystems of $\mathrm{\Phi }$. 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.