Eigenvalues of unipotent elements in cross-characteristic representations of finite classical groups.
This paper is a significant contribution to a long-term program intended to study minimum polynomials of group elements in (modular) irreducible representations \(\Theta\) of degree \(>1\). An important part of this program deals with elements \(g\), of prime-power order \(s\), of finite classical groups \(G\) defined over fields of characteristic \(p\), where the characteristic of the representation \(\Theta\) in question is coprime to \(p\). Assume in addition that \(g\) is semisimple (that is, \(s\) and \(p\) are coprime). In a previous paper [J. Algebra 243, No. 1, 228-263 (2001); corrigendum 296, No. 1, 249-252 (2006; Zbl 1017.20036)] the authors treated the case where \(g\) belongs to a parabolic subgroup of \(G\). In general, an optimal lower bound for the degree of the minimum polynomial of \(\Theta(g)\) has recently been proved by the second author and the reviewer [in Proc. Lond. Math. Soc. (3) 97, No. 3, 623-668 (2008; Zbl 1173.20012)]. The paper under review handles the case where \(p\mid s\), i.e. \(g\) is unipotent (and \(G\) is a quasi-simple classical group). One can summarize the main results of the paper as follows: either the spectrum of \(\Theta(g)\) consists of all \(s\)-roots of unity, or \((G,g,\Theta)\) belongs to a short list of explicitly described exceptions. The major kind of exceptions is that \(g^{s/p}\) is a transvection, \(\Theta\) is a Weil representation, and either \(G=\text{Sp}(2m,p)\) with \(p>2\), or \(G=\text{SU}(m,2)\). Other exceptions can occur only for some specific representations of \(\text{Sp}(4,p)\), \(\text{SU}(3,p)\), \(\text{SL}(2,p)\), \(\text{SL}(2,p^2)\), \(\text{Sp}(8,3)\), \(\text{Sp}(6,3)\), \(\text{Sp}(6,2)\), \(\text{Sp}(4,9)\), \(\text{SU}(7,2)\), \(\text{Sp}(5,2)\), and \(\text{SU}(4,3)\). The authors also prove a lower bound for the multiplicity of each eigenvalue of \(\Theta(g)\) when the rank of \(G\) is large enough. One should consult the paper for precise formulations of all these interesting results.
scientific article; zbMATH DE number 5277608
scientific article