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

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.

6 pages

References in corpus (2)

• Representation zeta functions of some compact p-adic analytic groups
• Representation growth and representation zeta functions of groups

Cited by in corpus (4)

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• Zeta functions associated to admissible representations of compact p-adic Lie groups
• Poincaré series of Lie lattices and representation zeta functions of arithmetic groups
• Triple mean values of Witten $L$-functions