Representation growth of compact linear groups

We study the representation growth of simple compact Lie groups and of ${\mathrm{S}\mathrm{L}}_n(\mathcal{O})$, where $\mathcal{O}$ is a compact discrete valuation ring, as well as the twist representation growth of ${\mathrm{G}\mathrm{L}}_n(\mathcal{O})$. This amounts to a study of the abscissae of convergence of the corresponding (twist) representation zeta functions. We determine the abscissae for a class of Mellin zeta functions which include the Witten zeta functions. As a special case, we obtain a new proof of the theorem of Larsen and Lubotzky that the abscissa of Witten zeta functions is $r\mathrm{/}\kappa$, where $r$ is the rank and $\kappa$ the number of positive roots. We then show that the twist zeta function of ${\mathrm{G}\mathrm{L}}_n(\mathcal{O})$ exists and has the same abscissa of convergence as the zeta function of ${\mathrm{S}\mathrm{L}}_n(\mathcal{O})$, provided $n$ does not divide $\text{char}\mathcal{O}$. We compute the twist zeta function of ${\mathrm{G}\mathrm{L}}_2(\mathcal{O})$ when the residue characteristic $p$ of $\mathcal{O}$ is odd, and approximate the zeta function when $p=2$ to deduce that the abscissa is $1$. Finally, we construct a large part of the representations of ${\mathrm{S}\mathrm{L}}_2({\mathbb{F}}_q[[t]])$, $q$ even, and deduce that its abscissa lies in the interval $[1,5\mathrm{/}2]$.