Extensions for Frobenius kernels.
Let \(G\) be a simply connected simple algebraic group defined and split over a prime field \(\mathbb{F}_p\). We follow the notations of \textit{J. C. Jantzen} [Representations of algebraic groups. (2nd ed. Mathematical Surveys and Monographs 107. Providence, RI: American Mathematical Society) (2003; Zbl 1034.20041)]. The representation theory of the Frobenius kernels \(G_r\) is important for the representation theory of both \(G\) and its finite subgroups \(G(\mathbb{F}_q)\). This paper gives precise information on several \(\text{Ext}\) groups of \(G_r\)-modules, including cases of small characteristic. This inclusiveness necessitates many case distinctions. The authors compute the first \(G_r\)-cohomology of each induced representation \(H^0(\lambda)\) by way of its first \(B_r\)-cohomology. Thanks to this effort we now know all \(H^1(G_r,H^0(\lambda))^{(-r)}\). Each of them has a good filtration. One may hope that the good filtration property holds for higher \(H^i\) too, as predicted by Donkin. Next they study the \(G_r\)-extensions between simple modules. Here the results are less definite. They express the \(G\)-module \(E=\text{Ext}^1_{G_r}(L(\lambda),L(\mu))\) in terms of \(\text{Ext}^1_G\) of suitable \(G\)-modules, but sometimes only as a vector space. Then new bounds are given for the weights that occur in \(E\) or in the auxiliary \(G\)-modules.
scientific article; zbMATH DE number 2055641
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