On the self-dual representations of finite groups of Lie type
The author presents various results on self-dual irreducible complex representations of a finite group \(G\) of Lie type, motivated in part by known related results in the theory of representations of semisimple algebraic groups. In the context of finite groups, the self-dual irreducible representations are those with real-valued character. The author concentrates on self-dual irreducible generic representations, which may be interpreted as constituents of the Gelfand-Graev representations of \(G\). Sufficient conditions are given in terms of the action of the centre of \(G\) for a self-dual irreducible generic representation of \(G\) to be of orthogonal type (equivalently, to be the character of a real representation of \(G\)). These conditions imitate the algebraic group theory. Perhaps the author's most novel result is the following. Let \(q\) be a power of a prime with \(q\equiv 3\bmod 4\). Then there exists an irreducible self-dual complex representation of the special linear group \(\text{SL}(6,q)\) which is of orthogonal type although its central involution acts non-trivially. Similarly, there exists an irreducible self-dual complex representation of the special linear group \(\text{SL}(6,q)\) which is of symplectic type although its central involution acts trivially (the representation may be viewed as faithful representation of the projective group \(\text{PSL}(6,q)\)). Such representations are uncommon and might be thought unlikely to exist (they do not exist in the case when \(q\equiv 1\bmod 4\)). However, it is now known that they also exist in \(\text{SL}(4n+2,q)\) when \(n\geq 1\) and \(q\equiv 3\bmod 4\).
self-dual irreducible complex representations
scientific article; zbMATH DE number 1247579