The representation zeta function of a FAb compact p-adic Lie group vanishes at -2

TL;DR

This paper investigates the behavior of the representation zeta function of FAb compact p-adic Lie groups at s = -2, establishing that it vanishes for infinite groups when p > 2, extending classical finite group results.

Contribution

It proves that the representation zeta function Z(G, -2) equals zero for infinite FAb compact p-adic Lie groups with p > 2, complementing known finite group results.

Findings

Z(G, -2) = 0 for infinite G when p > 2

Extension of finite group results to infinite p-adic Lie groups

Provides new insights into the structure of representation zeta functions

Abstract

Let G by compact p-adic Lie group and suppose that G is FAb, i.e., that H/[H,H] is finite for every open subgroup H of G. The representation zeta function Z(G,s) encodes the distribution of continuous irreducible complex characters of G. Here s denotes a complex variable and Z(G,s) is defined as the Dirichlet generating function whose nth coefficient is equal to the number of irreducible characters of G of degree n. For p greater than 2 it is known that Z(G,s) defines a meromorphic function on the complex plane. Wedderburn's structure theorem for semisimple algebras implies that ZG,-2) = |G| for finite G. We complement this classic result by proving that Z(G,-2) = 0 for infinite G, assuming that p is greater than 2.