Regular orbits of linear groups with an application to the \(k(GV)\)-problem. II
This paper completes the research begun by the author in the first paper of this series, [J. Algebra 227, No. 2, 395-432 (2000; see the preceding review Zbl 0970.20005)]. Let \(p\) be a prime. Let \(G\) be a finite group of \(p'\)-order and let \(V\) be a faithful \(FG\)-module of finite dimension \(n\), where \(F\) is the field with \(p\) elements. Suppose that \(G\) contains a quasi-simple normal subgroup \(H\) that acts irreducibly on \(V\) and that \(H/Z(H)\) is a group of Lie type. Suppose also that \(G\) has no regular orbit on \(V\). Then \(H/Z(H)\), \(n\) and \(p\) belong to a finite list of possibilities, described in our earlier review. The author works his way systematically through the different families of groups of Lie type. The greatest problems in his analysis arise for symplectic and unitary groups, and tend to be associated with the case that \(V\) affords a Weil representation of \(H\). A certain number of rather specific module actions have to be examined in detail for a small number of groups. The author also makes good use of a paper of \textit{G. M. Seitz} and \textit{A. E. Zalesskij}, [J. Algebra 158, No. 1, 233-243 (1993; Zbl 0789.20014)], which gives lower bounds for the dimension of any non-trivial projective representation of a group of Lie type.
scientific article; zbMATH DE number 1471027