Character values, Schur indices and character sheaves

Abstract

In this paper we are concerned with the problem of determining the character values and Schur indices of a finite group of Lie type over F q {\mathbb F}_q . We show that (under some conditions on q q ) these values lie in the ring of algebraic integers generated by ( 1 + ± q ) / 2 (1+ \sqrt {\pm q})/2 and roots of unity of order prime to q q . Furthermore, we determine the Schur indices for some of the (nonrational) unipotent characters in exceptional groups. Our results, combined with previous results due to Gow, Ohmori and Lusztig, imply that there are only 6 cases left where the Schur index of a cuspidal unipotent character remains unknown. Our methods rely, in an essential way, on Lusztig’s theory of character sheaves.