Irreducible modules and parabolic subgroups

SummaryIt seems desirable, from the point of view of finite group theorists, to develop further the representation theory of finite Chevalley groups in the characteristic of their definition (say, p) by focusing attention on the full array of p-local subgroups (that is, the parabolics) rather than the usual single Borel or Cartan subgroup.The following observation may be regarded as a generalization to arbitrary parabolic subgroups of the standard result [3, Theorem 39(d)] for a Borel subgroup, namely, that in an irreducible module, the subspace fixed by a maximal unipotent subgroup is 1-dimensional, and so affords an irreducible module for a Levi complement, which is an (abelian) Cartan subgroup. It seems natural to state the result first for algebraic groups