2014/08/13 by Friedlander, Eric M.
#20C20 #20G05 #20G10 #FOS: Mathematics #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.1408.2918
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 \mathbb 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".