### 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 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.

Original language | English (US) |
---|---|

Pages (from-to) | 327-390 |

Number of pages | 64 |

Journal | Journal of Algebra |

Volume | 271 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2004 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Journal of Algebra*,

*271*(1), 327-390. https://doi.org/10.1016/S0021-8693(03)00174-1

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

Research output: Contribution to journal › Article

*Journal of Algebra*, vol. 271, no. 1, pp. 327-390. https://doi.org/10.1016/S0021-8693(03)00174-1

}

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 -