Generic and maximal Jordan types

For a finite group scheme G over a field k of characteristic p>0, we associate new invariants to a finite dimensional kG-module M. Namely, for each generic point of the projectivized cohomological variety we exhibit a “generic Jordan type” of M. In the very special case in which G=E is an elementary abelian p-group, our construction specializes to the non-trivial observation that the Jordan type obtained by restricting M via a generic cyclic shifted subgroup does not depend upon a choice of generators for E. Furthermore, we construct the non-maximal support variety Γ(G) _M , a closed subset of which is proper even when the dimension of M is not divisible by p.