Unipotent elements of finite groups of Lie type and realization fields of their complex representations

• Published in: Journal of algebra 2004, Vol.271 (1), p.327-390
• Main Authors: Tiep, Pham Huu, Zalesskiı̆, A.E.
• Format: Article
• Language: English
• ISSN: 0021-8693; 1090-266X
• DOI: 10.1016/S0021-8693(03)00174-1

Let p be a prime. This paper classifies finite connected reductive groups G in characteristic p with the property that all complex character values of G belong to an unramified above p extension of the field of rational numbers. The main application of these results is to the problem of describing the irreducible complex (or p-adic) representations of G that remain absolutely irreducible under the Brauer reduction modulo p. An efficient approach to solve this problem for p>3 has been developed in our paper [Proc. London Math. Soc. (3) 84 (2002) 439]. Together with [Proc. London Math. Soc. (3) 84 (2002) 439], Theorem 1.9 of the paper solves this problem for many finite connected reductive groups in characteristic p>3. Additionally, we show that all complex representations of any finite connected reductive group with no composition factor of type E 7(2 f ), E 8(2 f ), and E 8(5 f ) can be realized over a quadratic extension of an unramified (above p) extension of  Q .