On representation zeta functions of groups and a conjecture of Larsen and Lubotzky

TL;DR

This paper investigates the properties of representation zeta functions for certain groups, proving functional equations for p-adic groups and addressing a conjecture on convergence for arithmetic groups under specific assumptions.

Contribution

It establishes functional equations for zeta functions of p-adic analytic groups and proves a conjecture on the abscissa of convergence for specific arithmetic groups, assuming Serre's conjecture.

Findings

Functional equations for zeta functions of p-adic groups

Proof of Larsen and Lubotzky's conjecture under certain assumptions

Connections between representation growth and number field properties

Abstract

We study zeta functions enumerating finite-dimensional irreducible complex linear representations of compact p-adic analytic and of arithmetic groups. Using methods from p-adic integration, we show that the zeta functions associated to certain p-adic analytic pro-p groups satisfy functional equations. We prove a conjecture of Larsen and Lubotzky regarding the abscissa of convergence of arithmetic groups of type A_2 defined over number fields, assuming a conjecture of Serre on lattices in semisimple groups of rank greater than 1.