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

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.