On the existence of mock injective modules for algebraic groups

Let G be an affine algebraic group scheme over an algebraically closed field k of characteristic p>0, and let G_r denote the rth Frobenius kernel of G. Motivated by recent work of Friedlander, the authors investigate the class of mock injective G-modules, which are defined to be those rational G-modules that are injective on restriction to Gr for all r≥1. In this paper, the authors provide necessary and sufficient conditions for the existence of non-injective mock injective G-modules, thereby answering a question raised by Friedlander. Furthermore, the authors investigate the existence of non-injective mock injectives with simple socles. Interesting cases are discovered that show that this can occur for reductive groups, but will not occur for their Borel subgroups.