Geometric Invariants of Representations of Finite Groups

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 $\tau$ 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 $\pi$-points for $k\tau$. More generally, we provide a construction of vector bundles for the semi-direct product $G\rtimes \tau$ of an infinitesimal group scheme $G$ and a finite group $\tau$. A major motivation for this study is to further our understanding of the relationship between representations of $\mathbb G(\mathbb F_p)$ 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.