Arithmetic Groups, Base Change, and Representation Growth

Abstract

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 $${\alpha(\mathbf{G}(O_S)) = \lim_{n \rightarrow\infty} \log R_n(\mathbf{G}(O_S)) / \log n}$$ 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 $${\alpha(\mathbf{G}(O_S))}$$ 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.