2019/06/16 by Friedlander, Eric M.
#20C20 #20G05 #20G10 #FOS: Mathematics #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.1906.06733
J. Pevtsova and the author constructed a ``universal p-nilpotent operator" for an infinitesimal group scheme G over a field k of characteristic p > 0 which led to coherent sheaves on the scheme of 1-parameter subgroups of G associated to a G-module M. Of special interest is the fact that these coherent sheaves are vector bundles if M is of constant Jordan type. In this paper, we provide similar invariants for a finite group τ which recover the invariants earlier obtained for elementary abelian p-groups. To do this, we replace the analogue of 1-parameter subgroups by a refined version of equivalence classes of π-points for kτ. More generally, we provide a construction of vector bundles for the semi-direct product G\rtimes τ of an infinitesimal group scheme G and a finite group τ. A major motivation for this study is to further our understanding of the relationship between representations of \mathbb G(\mathbb Fp) and \mathbb G(r) associated to a finite dimensional rational \mathbb G-module M, where \mathbb G is a reductive group with r-th Fobenius kernel \mathbb G(r). Using vector bundles, we extend and sharpen earlier results comparing support varieties.