Revisiting the Steinberg representation at arbitrary roots of 1

TL;DR

This paper extends fundamental results on quantum group representations at roots of unity to arbitrary orders, analyzing the Steinberg representation and establishing properties of projective and injective modules in this broader setting.

Contribution

It generalizes known results at odd roots of unity to arbitrary roots, providing new insights into the structure of quantum group representations and the Steinberg representation.

Findings

Category Rep(G_q) has enough projectives and injectives

A G_q-representation's projectivity/injectivity is equivalent to its restriction to the small quantum group

Results apply to arbitrary roots of unity, not just odd order q

Abstract

We consider quantum group representations for a semisimple algebraic group G at a complex root of unity q. Here q is allowed to be of any order. We revisit some fundamental results of Parshall-Wang and Andersen-Polo-Wen from the 90's. In particular, we show that the category Rep(G_q) of quantum group representations has enough projectives and injectives, and that a G_q-representation is projective (resp. injective) if and only if its restriction to the small quantum group is projective (resp. injective). Our results reduce to an analysis of the Steinberg representation in the simply-connected setting, and are well-known at odd order q via works of the aforementioned authors. The details at arbitrary q have, to our knowledge, not appeared in the literature up to this point.