Sheaf homology and complete reducibility
Let G denote a Chevalley group over finite field \(F_ q\) of characteristic p with a fixed Borel subgroup B. Consider an \(F_ qG\)- module V satiyfing \(V=<C_ V(O_ p(B))^ g:\) \(g\in G>\). Following the 0-th homology construction in \textit{M. A. Ronan} and \textit{S. D. Smith} [ibid. 96, 319-346 (1985; Zbl 0604.20043)] one can define a sheaf \({\mathcal F}_ V\) on V in terms of subspaces \(V_ 0\) of \(C_ V(O_ p(P_ 0))\) when \(P_ 0\) runs over the parabolics of G containing B. The problem of splitting of V with only two composition factors S and \(T\cong V/S\) reduces to the splitting of \({\mathcal F}_ V\) in the direct sum of \({\mathcal F}_ S\) and \({\mathcal F}_ T\) and some homological questions for \(H_ 0({\mathcal F}_ T)\). Then the author gives an elegant sufficient condition for complete reducibility of V: all composition factors of V are modules \(T_ i\) with minimal highest weights \(\lambda_ i\) not pairwise adjacent in the Dynkin diagram (if \(p=2\), G cannot be of type \(BC_ 3)\), or all \(T_ i\) are pairwise isomorphic.
scientific article; zbMATH DE number 3985506