PBW theorems and Frobenius structures for quantum matrices

Let G be either of Mat(n), GL(n) or SL(n), let O_q(G) be the quantum function algebra - over Z[q,q^{-1}] - associated to G, and let O_e(G) be the specialisation of O_q(G) at a root of unity, of odd order l. Then O_e(G) is a module over the corresponding classical function algebra O(G) via the quantum Frobenius morphism, which embeds O(G) as a central subbialgebra of O_e(G). In this note we prove a PBW-like theorem for O_q(G) - more or less known in literature, but not in this form (to the best of the author's knowledge) - and we show that it yields explicit bases of O_e(G) over O(G) when G is Mat(n) or GL(n): in particular, O_e(G) is free of rank l^{dim(G)}. Also, we apply the latter result to prove that O_e(G) is a free Frobenius extensions over O(G), and to compute explicitly the corresponding Nakayama automorphism, again for G being Mat(n) or GL(n) . This extends previous results by Brown, Gordon and Stroppel (see [BG], [BGS2]).Comment: AMS-TeX file, 10 pages. This paper is a natural evolution - now in *final version* - of preprint math.QA/0608016, which has been withdraw

Forced gradings and the Humphreys-Verma conjecture

Let $G$ be a semisimple, simply connected algebraic group defined and split over a prime field ${\mathbb F}_p$ of positive characteristic. For a positive integer $r$, let $G_r$ be the $r$th Frobenius kernel of $G$. Let $Q$ be a projective indecomposable (rational) $G_r$-module. The well-known Humprheys-Verma conjecture (cf. \cite{Ballard}) asserts that the $G_r$-action on $Q$ lifts to an rational action of $G$ on $Q$. For $p\geq 2h-2$ (where $h$ is the Coxeter number of $G$), this conjecture was proved by Jantzen in 1980, improving on early work of Ballard. However, it remains open for general characteristics. In this paper, the authors establish several graded analogues of the Humphreys-Verma conjecture, valid for all $p$. The most general of our results, proved in full here, was announced (without proof) in an earlier paper. Another result relates the Humphreys-Verma conjecture to earlier work of Alperin, Collins, and Sibley on finite group representation theory. A key idea in all formulations involves the notion of a forced grading. The latter goes back, in particular, to the recent work of the authors, relating graded structures and $p$-filtrations. The authors anticipate that the Humphreys-Verma conjecture results here will lead to extensions to smaller characteristics of these earlier papers