Representations and maximal subgroups of finite groups of Lie type
Let \(\bar G\) be a simple, simply connected algebraic group over K, the algebraic closure of \(F_ p\). Let \(\sigma\) be an endomorphism of \(\bar G\) such that the fixed point group \(G=\bar G_{\sigma}\) is finite and quasisimple. Let V be an irreducible, but not necessarily absolutely irreducible kG-module where k is either K or a finite subfield of K. The author proves several theorems concerning such modules. The first says that a Levi factor L of a parabolic subgroup of G acts irreducibly on V and has a canonical L-invariant complement. This result had been proved earlier by \textit{S. D. Smith} [J. Algebra 75, 286-289 (1982; Zbl 0496.20030)] for the case k a splitting field. The second theorem states that, except for four exceptions when V is the Steinberg module, V is imprimitive. Another result shows that it is possible to lift G-invariant bilinear and quadratic forms on V to \(\bar G\)-invariant forms on \(K\otimes V\). The final results concern fields of definition of representations of G. The results have applications to the determination of the maximal subgroups of finite classical groups. One wants to determine explicitly all triples \((X,Y,V)\) with \(X<Y<I(V)\), where I(V) is one of the classical groups associated with V, X is of Lie type and Y is quasisimple. The results in this paper together with the results of an earlier paper by the author [Mem. Am. Math. Soc. 365 (1987; Zbl 0624.20022)] go a long way toward the classification of such triples.
simple, simply connected algebraic group
scientific article; zbMATH DE number 4179640