Representation growth of linear groups

Abstract

Let $\Gamma$ be a group and $r_n(\Gamma )$ the number of its $n$-dimensional irreducible complex representations. We define and study the associated representation zeta function ${\mathcal{Z}}_{\Gamma }(s)={\sum }_{n=1}^{\infty }{r}_n(\Gamma )n^{-s}$. When $\Gamma$ is an arithmetic group satisfying the congruence subgroup property then ${\mathcal{Z}}_{\Gamma }(s)$ has an “Euler factorization”. The “factor at infinity” is sometimes called the “Witten zeta function” counting the rational representations of an algebraic group. For these we determine precisely the abscissa of convergence. The local factor at a finite place counts the finite representations of suitable open subgroups $U$ of the associated simple group $G$ over the associated local field $K$. Here we show a surprising dichotomy: if $G(K)$ is compact (i.e. $G$ anisotropic over $K$) the abscissa of convergence goes to 0 when $\dim G$ goes to infinity, but for isotropic groups it is bounded away from $0$. As a consequence, there is an unconditional positive lower bound for the abscissa for arbitrary finitely generated linear groups. We end with some observations and conjectures regarding the global abscissa.