Involutions acting on representations. (Q2491800)

This paper is mainly concerned with reinterpreting various known results about the irreducible self-dual representations of the general linear and special linear groups. The author makes special use of a result of \textit{D. Prasad}, [J. Algebra 210, No. 1, 298-310 (1998; Zbl 0922.20020)], which gives a sufficient condition to decide whether certain special types of irreducible self-dual complex representations are of orthogonal or symplectic type. He also uses a theorem of Klyachko (1983), which constructs a model for the irreducible complex representations of the finite general linear group \(\text{GL}(n,q)\). (A model is a representation which contains each irreducible representation exactly once.) The model constructed by Klyachko is the sum of representations induced by one-dimensional representations of a family of subgroups of the general linear group. It is relatively straightforward to show that each irreducible constituent of the model, and hence each irreducible character of \(\text{GL}(n,q)\), has Schur index 1 over the rationals. An identical conclusion had earlier been obtained by Zelevinsky (1981). The author uses Klyachko's theorem, together with the lemma of Prasad, to determine whether a given irreducible self-dual representation of the finite special linear group is of orthogonal or symplectic type (equivalently, he determines the Frobenius-Schur indicators of all the complex irreducible characters of the special linear group). A similar result had been obtained by the reviewer in 1981, using the ideas of Zelevinsky. The paper contains further results in this vein, including some relating to the self-dual representations of the general linear group of a local field.