Real representations of the finite orthogonal and symplectic groups of odd characteristic (Q1065149)

The author addresses the general question: which complex irreducible representations of a finite classical group can be realized over the real field? In an earlier paper [Proc. Lond. Math. Soc., III. Ser. 47, 493--506 (1983; Zbl 0551.20023)], he dealt with the general linear groups. Now he turns to the orthogonal and symplectic groups of odd characteristic. As before, the general method is to investigate in detail the \(\mathbb R\)-elementary subgroups. The main results are as follows. Theorem 1. Each complex irreducible character of an orthogonal group \(O(n,q)\) (\(q\) odd) is the character of a real representation. A real-valued complex irreducible character of \(\mathrm{Sp}(2n,q)\) (\(q\) odd) has Schur index 2 or 1 over \(\mathbb R\) according as it is faithful or not. Theorem 2. Every real-valued complex irreducible character of a special orthogonal group \(\mathrm{SO}(n,q)\) (\(q\) odd) is the character of a real representation. The author also proves that, when \(q\equiv 1\pmod 4\), the sum of the degrees of the irreducible complex characters of \(\mathrm{Sp}(2n,q)\) is \(q^{n(n+1)}\prod^{n}_{i=1}(q^ i+1)\); and he conjectures that this remains true when \(q\equiv 3\pmod 4\).