Paper detail

Arithmetic groups, base change, and representation growth

Consider an arithmetic group $\mathbf{G}(O_S)$, where $\mathbf{G}$ is an affine group scheme with connected, simply connected absolutely almost simple generic fiber, defined over the ring of $S$-integers $O_S$ of a number field $K$ with respect to a finite set of places $S$. For each $n \in \mathbb{N}$, let $R_n(\mathbf{G}(O_S))$ denote the number of irreducible complex representations of $\mathbf{G}(O_S)$ of dimension at most $n$. The degree of representation growth $α(\mathbf{G}(O_S)) = \lim_{n \rightarrow \infty} \log R_n(\mathbf{G}(O_S)) / \log n$ is finite if and only if $\mathbf{G}(O_S)$ has the weak Congruence Subgroup Property. We establish that for every $\mathbf{G}(O_S)$ with the weak Congruence Subgroup Property the invariant $α(\mathbf{G}(O_S))$ is already determined by the absolute root system of $\mathbf{G}$. To show this we demonstrate that the abscissae of convergence of the representation zeta functions of such groups are invariant under base extensions $K \subset L$. We deduce from our result a variant of a conjecture of Larsen and Lubotzky regarding the representation growth of irreducible lattices in higher rank semi-simple groups. In particular, this reduces Larsen and Lubotzky's conjecture to Serre's conjecture on the weak Congruence Subgroup Property, which it refines.