BGG categories in prime characteristics

TL;DR

This paper explores the structure of the BGG category for quantum groups at roots of unity in positive characteristic, establishing key theorems about simple modules, Verma modules, and tilting modules, with implications for classical cases.

Contribution

It introduces a Steinberg tensor product theorem for simple modules in the quantum BGG category at roots of unity in positive characteristic.

Findings

Proved a Steinberg tensor product theorem for simple modules.

Established a finite filtration and linkage principle for Verma modules.

Identified the special Verma module as simple and projective/injective.

Abstract

Let $\mathfrak g$ be a simple complex Lie algebra. In this paper we study the BGG category $\mathcal O_q$ for the quantum group $U_q(\mathfrak g)$ with $q$ being a root of unity in a field $K$ of characteristic $p >0$. We first consider the simple modules in $\mathcal O_q$ and prove a Steinberg tensor product theorem for them. This result reduces the problem of determining the corresponding irreducible characters to the same problem for a finite subset of finite dimensional simple modules. Then we investigate more closely the Verma modules in $\mathcal O_q$. Except for the special Verma module, which has highest weight $-\rho$, they all have infinite length. Nevertheless, we show that each Verma module has a certain finite filtration with an associated strong linkage principle. The special Verma module turns out to be both simple and projective/injective. This leads to a family of projective modules in $\mathcal O_q$, which are also tilting modules. We prove a reciprocity law, which gives a precise relation between the corresponding family of characters for indecomposable tilting modules and the family of characters of simple modules with antidominant highest weights. All these results are of particular interest when $q = 1$, and we have paid special attention to this case.