A universal leading-residue formula for Witten zeta functions

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 $\xi_\Phi(s)$ has a simple pole at $s=2/h$, with residue $\mathop{\rm Res}_{s=2/h}\xi_\Phi(s)=\frac{2(2\pi)^{r/2}\sqrt{\det C_\Phi}}{h|W|}\frac{\prod_{i=1}^{r-1}\Gamma(1-d_i/h)}{\Gamma(1-1/h)^r}$.

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 $X$.