Toward partial Verma functors of ${\mathcal{U}}^{[r]}(\mathfrak{g})$ and related results

Abstract

This note extends the Steinberg Tensor product theorem from the Frobenius kernel $G_{(r)}$ to the deformation ${\mathcal{U}}^{[r]}(\mathfrak{g})$ of its distribution algebra. As a Corollary we proof some conjectures from \cite{Wes}. Further it describes the graded representation theory of ${\mathcal{U}}^{[r]}(\mathfrak{g})$ at a generic semisimple p-central character as a twist of the category of graded $G_{(r-1)}$-modules. We conclude by explaining this relation for $S{L}_2$ explicitely as well as computing the center in this case.