Shifted generic cohomology. (Q2860721)

Let \(G\) be a simply connected, semisimple algebraic group over an algebraically closed field \(k\) of prime characteristic \(p\) with \(G\) being split over \(\mathbb F_p\). For each \(r>1\), associated to \(G\) is the finite group of Lie type \(G(\mathbb F_{p^r})\) which arises as the subgroup of \(\mathbb F_{p^r}\)-rational points in \(G\). For a rational \(G\)-module \(M\), in the seminal work of the first two authors with \textit{E. Cline} and \textit{W. van der Kallen} [Invent. Math. 39, 143-163 (1977; Zbl 0336.20036)], it was shown that, as \(r\) increases, the cohomology groups \(H^m(G(\mathbb F_{p^r}),M)\) stabilize. This stable limit is known as the \textit{generic} \(G\)-cohomology of \(M\), denoted \(H^m_{\text{gen}}(G,M)\). It was further shown therein that this generic cohomology could be identified with the \(G\)-cohomology of a possible Frobenius twist of the module \(M\). The simple \(G(\mathbb F_{p^r})\)-modules arise as the restrictions of the simple rational \(G\)-modules which have \(p^r\)-restricted highest weight, and the goal of the paper under review is to more directly relate the \(G(\mathbb F_{p^r})\)-cohomology of a simple module to the \(G\)-cohomology of a simple (untwisted) \(G\)-module. Let \(L(\lambda)\) denote such a simple module where \(\lambda\) denotes the highest weight.NEWLINENEWLINE The main result is that, for sufficiently large \(r\), depending on the underlying root system and the cohomological degree, for any \(p^r\)-restricted weight \(\lambda\), there is another \(p^r\)-restricted weight \(\lambda'\) such that \(H^m(G(\mathbb F_{p^r}),L(\lambda))\cong H^m_{\text{gen}}(G,L(\lambda'))\cong H^m(G,L(\lambda'))\). Here the weight \(\lambda'\) is determined by \(\lambda\) and referred to as a \(q\)-shift of \(\lambda\) (where \(p^r\) is denoted by \(q\) in the paper). The authors work in the more general setting of extensions between simple \(G(\mathbb F_{p^r})\)-modules (and obtain an analogous result in that setting under the constraint that the prime \(p\) is somewhat ``large''). The main result is obtained through the interplay of cohomology (or extensions) over \(G(\mathbb F_{p^r})\), \(G\), and the \(r\)-th Frobnenius kernel \(G_r\) of \(G\). \(G(\mathbb F_{p^r})\)-extensions are related to \(G\)-extensions via the induced module \(\text{ind}_{G(\mathbb F_{p^r})}^Gk\) which has a useful filtration by tensor products of standard induced modules that was introduced by this reviewer with \textit{D. Nakano} and \textit{C. Pillen} [Int. Math. Res. Not. 2012, No. 12, 2817-2866 (2012; Zbl 1253.20046)]. Using a Lyndon-Hochschild-Serre spectral sequence, one can relate \(G\)-extensions to \(G_r\)-extensions for the \(r\)-th Frobenius kernel \(G_r\) of \(G\). Certain bounds on weights of \(G_r\)-extensions then come into play.NEWLINENEWLINE In an important consequence of the main result, the authors show that there is a constant \(C\), depending on \(m\) and the underlying root system, which bounds \(H^m(G(\mathbb F_{p^r}),L)\) for any simple module \(L\). In the process of proving the main result, the authors also prove an interesting result on the extent to which the \(p\)-adic expansions of weights \(\lambda\), \(\mu\) can differ and still have \(\text{Ext}^m_{G(\mathbb F_{p^r})}(L(\lambda),L(\mu))\) be non-zero. As a final note, the results in this paper were inspired in part by the work and questions posed by the third author [in Commun. Algebra 40, No. 12, 4702-4716 (2012; Zbl 1269.20041)].