On the vanishing ranges for the cohomology of finite groups of Lie type. (Q2895806)

Let \(G\) be simple algebraic group defined and split over the prime field \(\mathbb F_p\) with \(p>0\). Let \(q=p^r\) and let \(k\) be an algebraic closure of \(\mathbb F_q\). We are interested in the first nontrivial \(H^i(G(\mathbb F_q),k)\) with \(i>0\). We want to know or estimate in what degree it occurs. And we would like to locate a nontrivial cohomology class in this minimal degree.NEWLINENEWLINE For example, Theorem 5.2 in the paper reads: Suppose that \(\Phi\) is of type \(C_n\) with \(p>2n\). Then (a) \(H^i(G(\mathbb F_q),k)=0\) for \(0<i<r(p-2)\); (b) \(H^{r(p-2)}(G(\mathbb F_q),k)\cong k\).NEWLINENEWLINE The authors start with using the Lang map \(L:G/G(\mathbb F_q)\to G\) to recast the \(G\)-module structure on \(\mathcal G_r(k):=\text{ind}^G_{G(\mathbb F_q)}k\) in terms of the \(G\times G\)-module structure on \(k[G]\). This is then used to establish a filtration of \(\mathcal G_r(k)\) with factors of the form \(H^0(\lambda)\otimes H^0(\lambda^*)^{(r)}\), one for each dominant \(\lambda\). Here \((r)\) denotes a Frobenius twist, \(\lambda^*=-w_0\lambda\), and \(H^0(\lambda)\) is the costandard module \(\nabla(\lambda)\) with highest weight \(\lambda\). Next they study the \(H^i(G,H^0(\lambda)\otimes H^0(\lambda^*)^{(r)})\) through Hochschild-Serre spectral sequences of the form \(E_2^{pq}=H^p(G/G_1,H^q(G_1,M))\Rightarrow H^{p+q}(G,M)\) with \(M\) a \(G\)-module and \(G_1\) the first Frobenius kernel. It is known that \(H^i(G_1,H^0(\lambda))\) involves the Kostant partition function and the authors finally get their results by means of an extensive study of root combinatorics.