Constructions for infinitesimal group schemes

Abstract:

Let $G$ be an infinitesimal group scheme over a field $k$ of characteristic $p >0$. We introduce the global $p$-nilpotent operator $\Theta _G: k[G] \to k[V(G)]$, where $V(G )$ is the scheme which represents 1-parameter subgroups of $G$. This operator $\Theta _G$ applied to $M$ encodes the local Jordan type of $M$ and leads to computational insights into the representation theory of $G$. For certain $kG$-modules $M$ (including those of constant Jordan type), we employ $\Theta _G$ to associate various algebraic vector bundles on $\mathbb {P}(G)$, the projectivization of $V(G)$. These vector bundles not only distinguish certain representations with the same local Jordan type, but also provide a method of constructing algebraic vector bundles on $\mathbb {P}(G)$.