For a given group \(G\), representation growth is the study of the asymptotic properties of the sequence \(r_i(G)\) of the number of irreducible (complex) representations of \(G\) of dimension \(i\). One of the main tools in this study is the representation zeta function of \(G\), which is the Dirichlet series associated to the sequence \(r_i(G)\). For many \(G\) of interest, including certain arithmetic groups, simple compact Lie groups and certain compact \(p\)-adic groups, this makes
sense, as \(r_i(G)\) will then be finite. While many things can be said about the compact \(p\)-adic case, this talk will focus on the simple Lie groups case, where the zeta functions are known as Witten zeta functions.
In a seminal paper, M. Larsen and A. Lubotzky developed the systematic study of representation zeta functions and emphasised the abscissa of convergence as the primary object of study. This is the real number which defines the half-plane of convergence of the zeta function and also gives the rate of polynomial growth of \(r_i(G)\). Larsen and Lubotzky proved that for a simple compact Lie group the abscissa of its representation zeta function is \(r/k\), where \(r\) is the rank and \(k\) the number of positive roots. Using examples, I will outline some of the ideas in a new proof of this result, and explain how it leads to a generalisation beyond Witten zeta functions and root systems. In particular, this goes some way towards explaining where the constant \(r/k\) comes from.
This is joint work with J. Häsä.