A note on Cabanes' decompositions for rational modules (Q1087991)

Let G be a simply connected semisimple algebraic group over an algebraically closed field which is split and defined over the prime field. Fix a pair P, \(P^-\) of opposite parabolic subgroups with unipotent radicals \(P_ u\) and \(P^-_ u\) and with Levi complement \(L_ P\). The author gives a proof (from ''first principles'') of a recent result of M. Cabanes showing that any simple rational G-module M admits a decomposition: \(M=M^{P_ u}\oplus (1-P^-_ u)M\) where the summands are \(L_ P\)-modules with \(M^{P_ u}\) simple. Moreover, the author completely gives the class of all finite dimensional rational G-modules that admit the above type of decomposition.