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

Pham Huu Tiep, A. E. Zalesskiǐ

Research output: Contribution to journalArticle

20 Citations (Scopus)

Abstract

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 E7 (2f), E8 (2f), and E8 (5f) can be realized over a quadratic extension of an unramified (above p) extension of ℚ.

Original languageEnglish (US)
Pages (from-to)327-390
Number of pages64
JournalJournal of Algebra
Volume271
Issue number1
DOIs
StatePublished - Jan 1 2004
Externally publishedYes

Fingerprint

Finite Groups of Lie Type
Reductive Group
P-adic
Modulo
Classify
Theorem

ASJC Scopus subject areas

  • Algebra and Number Theory

Cite this

Unipotent elements of finite groups of Lie type and realization fields their complex representations. / Tiep, Pham Huu; Zalesskiǐ, A. E.

In: Journal of Algebra, Vol. 271, No. 1, 01.01.2004, p. 327-390.

Research output: Contribution to journalArticle

@article{149ebafc487d42da95fd26c2566e4210,
title = "Unipotent elements of finite groups of Lie type and realization fields their complex representations",
abstract = "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 E7 (2f), E8 (2f), and E8 (5f) can be realized over a quadratic extension of an unramified (above p) extension of ℚ.",
author = "Tiep, {Pham Huu} and Zalesskiǐ, {A. E.}",
year = "2004",
month = "1",
day = "1",
doi = "10.1016/S0021-8693(03)00174-1",
language = "English (US)",
volume = "271",
pages = "327--390",
journal = "Journal of Algebra",
issn = "0021-8693",
publisher = "Academic Press Inc.",
number = "1",

}

TY - JOUR

T1 - Unipotent elements of finite groups of Lie type and realization fields their complex representations

AU - Tiep, Pham Huu

AU - Zalesskiǐ, A. E.

PY - 2004/1/1

Y1 - 2004/1/1

N2 - 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 E7 (2f), E8 (2f), and E8 (5f) can be realized over a quadratic extension of an unramified (above p) extension of ℚ.

AB - 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 E7 (2f), E8 (2f), and E8 (5f) can be realized over a quadratic extension of an unramified (above p) extension of ℚ.

UR - http://www.scopus.com/inward/record.url?scp=1642544093&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=1642544093&partnerID=8YFLogxK

U2 - 10.1016/S0021-8693(03)00174-1

DO - 10.1016/S0021-8693(03)00174-1

M3 - Article

AN - SCOPUS:1642544093

VL - 271

SP - 327

EP - 390

JO - Journal of Algebra

JF - Journal of Algebra

SN - 0021-8693

IS - 1

ER -