Twisted Signs for Finite General Linear Groups

Abstract

For an irreducible representation of a finite group G, a twisted sign is a generalization of the Frobenius-Schur indicator, which indicates whether a bilinear form preserving the G-action is symmetric or skew-symmetric. We will look at such signs for the finite general linear groups twisted by the transpose-inverse map. In particular, we will look at the relationship between these twisted signs and Harish-Chandra's philosophy of cusp forms. From this relationship, we show that these signs must all be one, recovering a result by Gow.