Second cohomology groups for Frobenius kernels and related structures.
Let \(p>2\) be a prime, \(G\) a simple simply connected affine algebraic group over an algebraically closed field \(k\) of characteristic \(p\), and let \(B\) be a Borel subgroup of \(G\). For any dominant weight \(\lambda\), consider the corresponding one-dimensional \(B\)-module \(k_\lambda\) and the induced \(G\)-module \(H^0(\lambda)=\text{Ind}^G_B(k_\lambda)\). The main result of the paper is the computation of the second cohomology group \(H^2(G_r,H^0(\lambda))\) for any Frobenius kernel \(G_r\). Previously, the \(G_1\)-cohomology of \(H^0(\lambda)\) was determined by \textit{H. H. Andersen} and \textit{J. C. Jantzen} [in Math. Ann. 269, 487-525 (1984; Zbl 0529.20027)], provided that \(p\) is larger than the Coxeter number. Furthermore, the first cohomology group \(H^1(G_r,H^0(\lambda))\) was computed by \textit{J. C. Jantzen} [in Prog. Math. 95, 289-315 (1991; Zbl 0749.17020)] for \(r=1\), and by the authors of the present paper [in J. Algebra 272, No. 2, 476-511 (2004; Zbl 1044.20031)] for all \(r\). As another important result, the authors also compute \(H^2(B,k_\lambda)\) for all \(\lambda\).
scientific article; zbMATH DE number 5123970
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