Reducibility modulo $p$ of complex representations of finite groups of Lie type: Asymptotical result and small characteristic cases

Abstract

Let G be a finite group of Lie type in characteristic p. This paper addresses the problem of describing the irreducible complex (or p-adic) representations of G that remain absolutely irreducible under the Brauer reduction modulo p. An efficient approach to solve this problem for p > 3 has been elaborated in earlier papers by the authors. In this paper, we use arithmetical properties of character degrees to solve this problem for the groups G ∈ { 2 B 2 (q), 2 G 2 (q), G 2 (q), 2 F 4 (q), F 4 (q), 3 D 4 (q)} provided that p < 3. We also prove an asymptotical result, which solves the problem for all finite groups of Lie type over F q with q large enough.