Unipotent elements in representations of finite groups of Lie type. (Q2885399)

A (finite dimensional) linear transformation is called a pseudo-reflection if it is diagonalizable with exactly two eigenvalues, one of which has multiplicity \(1\). A complete classification of linear groups generated by pseudo-reflections was given independently by A. Wagner and by A. E. Zalesskii and V. N. Serezhkin (see [\textit{A.~E. Zalesskij} and \textit{V.~N. Serezhkin}, Izv. Akad. Nauk SSSR, Ser. Mat. 44, 1279-1307 (1980; Zbl 0454.20047); translation in Math. USSR, Izv. 17, 477-503 (1981)] and references there). The present paper is an attempt to generalize these results.NEWLINENEWLINE The authors define a matrix to be almost cyclic if it is diagonalizable with at most one eigenvalue of multiplicity greater than \(1\), and they consider the question of classifying irreducible linear groups generated by almost cyclic elements. As a first step they prove the following theorem.NEWLINENEWLINE Suppose that \(H\) is a finite group which contains a normal subgroup \(G\) where \(G\) is a finite quasi-simple group of Lie type of defining characteristic \(r>2\), and \(H=\langle h,G\rangle\) where \(h\) is an \(r\)-element which does not centralize \(G\). Let \(F\) be an algebraically closed field with \(\text{char\,}F\neq r\), and let \(\varphi\) be an irreducible \(F\)-representation of \(H\) which is nontrivial on \(G\). Then \(\varphi(h)\) is almost cyclic if and only if \(H=GZ(H)\) and one of the following occurs: (1) \(G=\text{SL}(2,r)\) or \(\text{PSL}(2,r)\) and \(r>3\); (2) \(G=\text{SL}(2,9)\) or \(\text{PSL}(2,9)\) and \(\dim\varphi=4\) or \(5\); (3) \(G=\text{Sp}(4,3)\) or \(\text{PSp}(4,3)\) and \(\dim\varphi=4\), \(5\) or \(6\) (more precise details are given in cases (2) and (3)). -- A crucial role in the proof is played by an analysis of representations of cyclic extensions of extraspecial groups.