Revisiting the Steinberg representation at arbitrary roots of 1

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.