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 $\alpha (\mathbf{G}(O_S))={\lim }_{n\to \infty }\log R_n(\mathbf{G}(O_S))/\log n$ 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.

v1 Preliminary version. Comments are very welcome. v2 numerous corrections. v3 substantial revision. v4 final version, to appear in GAFA

References in corpus (2)

• Representation zeta functions of compact p-adic analytic groups and arithmetic groups
• Developments on the congruence subgroup problem after the work of Bass, Milnor and Serre

Cited by in corpus (1)

• Representation growth and representation zeta functions of groups