Asymptotics of the powers in finite reductive groups

Abstract

Let 𝐺 be a connected reductive group defined over Fq. Fix an integer M≥2, and consider the power map x↦xM on 𝐺. We denote the image of G(Fq) under this map by G(Fq)M and estimate what proportion of regular semisimple, semisimple and regular elements of G(Fq) it contains. We prove that, as q→∞, the set of limits for each of these proportions is the same and provide a formula. This generalizes the well-known results for M=1 where all the limits take the same value 1. We also compute this more explicitly for the groups GL(n,q) and U(n,q) and show that the set of limits are the same for these two group, in fact, in bijection under q↦−q for a fixed 𝑀.