Asymptotics of the powers in finite reductive groups

Abstract

Let 𝐺 be a connected reductive group defined over F q . Fix an integer M ≥ 2 , and consider the power map x ↦ x M on 𝐺. We denote the image of G ( F q ) under this map by G ( F q ) M and estimate what proportion of regular semisimple, semisimple and regular elements of G ( F q ) 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.