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Representation Theory

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Character values, Schur indices and character sheaves

Author: Meinolf Geck
Journal: Represent. Theory 7 (2003), 19-55
MSC (2000): Primary 20C15; Secondary 20G40
Published electronically: February 19, 2003
MathSciNet review: 1973366
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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 ${\mathbb F}_q$. We show that (under some conditions on $q$) these values lie in the ring of algebraic integers generated by $(1+ \sqrt {\pm q})/2$ and roots of unity of order prime to $q$. 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.

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Additional Information

Meinolf Geck
Affiliation: Institut Girard Desargues, bat. Jean Braconnier, Université Lyon 1, 21 av Claude Bernard, F–69622 Villeurbanne Cedex, France
MR Author ID: 272405

Received by editor(s): July 15, 2002
Received by editor(s) in revised form: November 13, 2002
Published electronically: February 19, 2003
Article copyright: © Copyright 2003 American Mathematical Society