Representation zeta functions of compact p-adic analytic groups and arithmetic groups

Abstract

We introduce new methods from $\mathfrak{p}$-adic integration into the study of representation zeta functions associated to compact $p$-adic analytic groups and arithmetic groups. They allow us to establish that the representation zeta functions of generic members of families of $p$-adic analytic pro-$p$ groups obtained from a global, “perfect” Lie lattice satisfy functional equations. In the case of “semisimple” compact $p$-adic analytic groups, we exhibit a link between the relevant $\mathfrak{p}$-adic integrals and a natural filtration of the locus of irregular elements in the associated semisimple Lie algebra, defined by the centralizer dimension.

Based on this algebro-geometric description, we compute explicit formulas for the representation zeta functions of principal congruence subgroups of the groups ${SL}_3\left(\mathfrak{o}\right)$, where $\mathfrak{o}$ is a compact discrete valuation ring of characteristic $0$, and of the groups ${SU}_3(\mathfrak{O},\mathfrak{o})$, where $\mathfrak{O}$ is an unramified quadratic extension of $\mathfrak{o}$. These formulas, combined with approximative Clifford theory, allow us to determine the abscissae of convergence of representation zeta functions associated to arithmetic subgroups of algebraic groups of type $A_2$. Assuming a conjecture of Serre on the congruence subgroup problem, we thereby prove a conjecture of Larsen and Lubotzky on lattices in higher-rank semisimple groups for algebraic groups of type $A_2$ defined over number fields.