Duality, central characters, and real-valued characters of finite groups of Lie type

Abstract

We prove that the duality operator preserves the Frobenius‐Schur indicators of characters of connected reductive groups of Lie type with connected center. This allows us to extend a result of D. Prasad which relates the Frobenius‐Schur indicator of a regular real-valued character to its central char acter. We apply these results to compute the Frobenius‐Schur indicators of certain real-valued, irreducible, Frobenius-invariant Deligne‐Lusztig characters, and the Frobenius‐Schur indicators of real-valued regular and semisimple characters of finite u nitary groups. Given a finite group G, and an irreducible finite dimensional complex representation ( , V ) of G with character , it is a natural question to ask what smallest field extension of is necessary to define a matrix representation correspondin g to ( , V ). If ( ) is the smallest extension of containing the values of , then the Schur index of over may be defined to be the smallest degree of an extension of ( ) over which ( , V ) may be defined. One may also consider the Schur index of over , which is 1 if ( , V ) may be defined over ( ), and 2 if it is not. If is a real-valued character, then the Schur index of over indicates whether ( , V ) may be defined over the real numbers. The Brauer‐Speiser Theorem states that if is a real-valued character, then the Schur index of over is either 1 or 2, and if the Schur index of over is 2, then the Schur index of over is 2. As finite groups of Lie type are of fundamental importance in t he theory of finite groups, it is of interest to understand the Schur indices ove r of their complex representations. In work of Gow and Ohmori [10, 11, 12, 19, 20], the Schur indices over of many irreducible characters of finite classical groups ar e determined. For the characters which are not covered by methods of Gow and Ohmori, it seems that the computation of the Schur index over is significantly more difficult. One example is that of the special linear group, which was completed by Turull [24]. Through methods developed by Lusztig, Geck, and Ohmori, the Schur index of unipotent characters have