A criterion for irreducibility of parabolic baby verma modules of reductive Lie algebras

Abstract Let G be a connected, reductive algebraic group over an algebraically closed field k of prime characteristic p and g = Lie ( G ) . In this paper, we study representations of g with a p-character χ of standard Levi form. When g is of type A n , B n , C n or D n , a sufficient condition for the irreducibility of standard parabolic baby Verma g -modules is obtained. This partially answers a question raised by Friedlander and Parshall in [Friedlander E. M. and Parshall B. J., Deformations of Lie algebra representations, Amer. J. Math. 112 (1990), 375-395]. Moreover, as an application, in the special case that g is of type A n or B n , and χ lies in the sub-regular nilpotent orbit, we recover a result of Jantzen in [Jantzen J. C., Subregular nilpotent representations of s l n and s o 2 n + 1 , Math. Proc. Cambridge Philos. Soc. 126 (1999), 223-257].