A universal leading-residue formula for Witten zeta functions

Let $\mathrm{\Phi }$ be an irreducible crystallographic root system of rank $r$, with Coxeter number $h$, Weyl group $W$, Cartan matrix $C_{\mathrm{\Phi }}$, and invariant degrees $2=d_1\le \cdots \le d_r=h$. We prove that Au’s normalized Witten zeta function ${\xi }_{\mathrm{\Phi }}(s)$ has a simple pole at $s=2\mathrm{/}h$, with residue ${}_{s=2\mathrm{/}h}{\xi }_{\mathrm{\Phi }}(s)=\frac{2(2\pi {)}^{r\mathrm{/}2}\sqrt{\det C_{\mathrm{\Phi }}}}{h\mathrm{\mid }W\mathrm{\mid }}\frac{{\prod }_{i=1}^{r-1}\mathrm{\Gamma }(1-d_i\mathrm{/}h)}{\mathrm{\Gamma }(1-1\mathrm{/}h{)}^r}$. 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 $X$.