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

TL;DR

This paper extends the Steinberg Tensor product theorem to a deformation of the distribution algebra of a Lie algebra, explores graded representation theory at generic characters, and provides explicit examples for SL2.

Contribution

It introduces partial Verma functors for ${ m U}^{[r]}(rak{g})$, extending classical theorems and describing the graded representation theory at generic p-central characters.

Findings

Extended Steinberg Tensor product theorem to ${ m U}^{[r]}(rak{g})$

Described graded representation theory at generic semisimple p-central characters

Computed the center explicitly for $SL_2$ case

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 $SL_2$ explicitely as well as computing the center in this case.