A universal leading-residue formula for Witten zeta functions

Abstract

Let $Φ$ be an irreducible crystallographic root system of rank $r$, with Coxeter number $h$, Weyl group $W$, Cartan matrix $C_Φ$, and invariant degrees $2=d_1\leq\cdots\leq d_r=h$. We prove that Au's normalized Witten zeta function $ξ_Φ(s)$ has a simple pole at $s=2/h$, with residue $\mathop{\rm Res}_{s=2/h}ξ_Φ(s)=\frac{2(2π)^{r/2}\sqrt{\det C_Φ}}{h|W|}\frac{\prod_{i=1}^{r-1}Γ(1-d_i/h)}{Γ(1-1/h)^r}$. The proof identifies the leading lattice coefficient with a convergent spherical Coxeter-discriminant integral at the critical exponent and evaluates this integral using the boundary pole of the Macdonald--Mehta--Opdam identity. Proper parabolic strata are shown to be strictly subcritical. This establishes Au's gamma-product-shape conjecture and his prediction in type $A_4$. We also obtain a direct, non-Tauberian asymptotic, with an explicit constant for every simple type, for the number of irreducible representations of dimension at most $X$.