Representation growth of compact linear groups

TL;DR

This paper investigates the growth rates of representations in compact Lie groups and related algebraic structures, determining convergence properties of associated zeta functions and providing new proofs and explicit calculations.

Contribution

It introduces new methods to analyze representation zeta functions of compact groups, including a novel proof of a known theorem and explicit computations for specific groups.

Findings

The abscissa of convergence for Witten zeta functions is r/κ.

The twist zeta function of GL_n(O) has the same abscissa as SL_n(O) when n does not divide char(O).

The abscissa for SL_2(F_q[[t]]) lies between 1 and 2.5.

Abstract

We study the representation growth of simple compact Lie groups and of $\mathrm{SL}_n(\mathcal{O})$, where $\mathcal{O}$ is a compact discrete valuation ring, as well as the twist representation growth of $\mathrm{GL}_n(\mathcal{O})$. 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 $r/\kappa$, where $r$ is the rank and $\kappa$ the number of positive roots. We then show that the twist zeta function of $\mathrm{GL}_n(\mathcal{O})$ exists and has the same abscissa of convergence as the zeta function of $\mathrm{SL}_n(\mathcal{O})$, provided $n$ does not divide $\text{char}\,{\mathcal{O}}$. We compute the twist zeta function of $\mathrm{GL}_2(\mathcal{O})$ 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 $\mathrm{SL}_2(\mathbb{F}_q[[t]])$, $q$ even, and deduce that its abscissa lies in the interval $[1,\,5/2]$.