Asymptotics of the powers in finite reductive groups

Abstract

Abstract Let 𝐺 be a connected reductive group defined over F q \mathbb{F}_{q} . Fix an integer M ≥ 2 M\geq 2 , and consider the power map x ↦ x M x\mapsto x^{M} on 𝐺. We denote the image of G ⁢ ( F q ) G(\mathbb{F}_{q}) under this map by G ⁢ ( F q ) M G(\mathbb{F}_{q})^{M} and estimate what proportion of regular semisimple, semisimple and regular elements of G ⁢ ( F q ) G(\mathbb{F}_{q}) it contains. We prove that, as q → ∞ q\to\infty , 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 M=1 where all the limits take the same value 1. We also compute this more explicitly for the groups GL ⁢ ( n , q ) \mathrm{GL}(n,q) and U ⁢ ( n , q ) \mathrm{U}(n,q) and show that the set of limits are the same for these two group, in fact, in bijection under q ↦ - q q\mapsto-q for a fixed 𝑀.