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 $2=d_1\le \cdots \le d_r=h$. 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.