Irreducible modules and Levi supplements (Q799807)

The purpose of the paper is to give an alternative proof of the following theorem of \textit{S. Smith} [J. Algebra 75, 286-289 (1982; Zbl 0496.20030)]. Let G be either a reductive group over an algebraically closed field k or a finite group with a finite saturated split BN-pair of characteristic p and k an algebraically closed field of the same characteristic. If P is a parabolic subgroup (of G) with unipotent radical U and Levi complement L, and M is a finite dimensional rational irreducible kG-module, then the fixed-point subspace \(M^ U\) is an irreducible kL-module. The proof treats simultaneously the algebraic group case and the finite split BN-pair case.