On injective modules for infinitesimal algebraic groups. II
Part I is a joint paper by the author, \textit{B. J. Parshall} and \textit{L. L. Scott} [J. Lond. Math. Soc., II. Ser. 31, 277-291 (1985; Zbl 0567.20024)]. Part II is not a direct continuation, though it deals with some of the same themes. When \(G\) is a semisimple algebraic group, defined and split over the prime field of an algebraically closed field of prime characteristic, the representation theory of \(G\) is intimately related to that of its (nonreduced) subgroup schemes \(TG_ r\), where \(T\) is a maximal (split) torus and \(G_ r\) is the kernel of the \(r\)th power of the Frobenius endomorphism. The author's aim is to study the structure of injective \(TG_ r\)-modules using parabolic subgroup schemes \(TP_ r\) and their Levi factors \(TL_ r\). Besides the usual restriction functors, one has ``lifting'' functors from the category of rational \(TL_ r\)- modules to the category of rational \(TG_ r\)-modules, the composite of inflation to \(TP_ r\) (or its opposite) and induction from \(TP_ r\) to \(TG_ r\). In terms of these functors, the author establishes isomorphisms between certain \(\text{Ext}^ n\) groups for \(TL_ r\) and \(TG_ r\). This applies especially to induced modules. He also studies the relationship between Jantzen's translation functors (for Levi factors) and the lifting functors, showing that lifting often commutes with translation.
scientific article; zbMATH DE number 5934
scientific article