Filtrations, 1-parameter subgroups, and rational injectivity

We investigate rational G-modules M for a linear algebraic group G over an algebraically closed field k of characteristic p>0 using filtrations by sub-coalgebras of the coordinate algebra k[G] of G. Even in the special case of the additive group Ga, interesting structures and examples are revealed. The “degree” filtration we consider for unipotent algebraic groups leads to a “filtration by exponential degree” applicable to rational G modules for any linear algebraic group G of exponential type; this filtration is defined in terms of 1-parameter subgroups and is related to support varieties introduced recently by the author for such rational G-modules. We formulate in terms of this filtration a necessary and sufficient condition for rational injectivity for rational G-modules. Our investigation leads to the consideration of two new classes of rational G-modules: those that are “mock injective” and those that are “mock trivial”.