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

✍ Scribed by Pham Huu Tiep; A.E. Zalesskiı̆

• Publisher: Elsevier Science
• Year: 2004
• Tongue: English
• Weight: 720 KB
• Volume: 271
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1016/s0021-8693(03)00174-1

✦ Synopsis

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.