A universal leading-residue formula for Witten zeta functions

Abstract

Let $\Phi$ be an irreducible crystallographic root system of rank $r$, with Coxeter number $h$, Weyl group $W$, Cartan matrix $C_\Phi$, and invariant degrees $2=d_1\leq\cdots\leq d_r=h$. We prove that Au's normalized Witten zeta function has a simple pole at $2/h$ and evaluate its residue as $\frac{2(2\pi)^{r/2}\sqrt{\det C_\Phi}}{h|W|}\cdot\frac{\prod_{i<r}\Gamma(1-d_i/h)}{\Gamma(1-1/h)^r}$. The central step evaluates the critical chamber integral in gamma values from the boundary pole of the Macdonald-Mehta-Opdam identity; two preparatory sections pass from the dominant-weight lattice to that convergent integral. This proves Au's conjecture on algebraic multiples of products of gamma values at rational arguments, including his $A_4$ prediction.