Polynomially and infinitesimally injective modules

The injective polynomial modules for a general linear group G of degree n are labelled by the partitions with at most n parts. Working over an algebraically closed field of characteristic p, we consider the question of which partitions correspond to polynomially injective modules that are also injective as modules for the restricted enveloping algebra of the Lie algebra of G. The question is related to the “index of divisibility” of a polynomial module in general, and an explicit answer is given for n=2.