### Abstract

In Külshammer's original question, H is replaced by a finite group and U by a Sylow p-subgroup of H; the answer is then known to be no in general. We obtain some results in the general case when H is non-connected and has positive dimension. Along the way, we prove existence and conjugacy results for maximal unipotent subgroups of non-connected linear algebraic groups. When G is reductive, we formulate Külshammer 's question and related conjugacy problems in terms of the nonabelian 1-cohomology of unipotent radicals of parabolic subgroups of G, and we give some applications of this cohomological approach. In particular, we analyse the case when G is a semisimple group of rank 2.

Original language | English |
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Pages (from-to) | 164-198 |

Number of pages | 35 |

Journal | Journal of Algebra |

Volume | 497 |

Early online date | 2 Oct 2017 |

DOIs | |

Publication status | Published - 1 Mar 2018 |

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### Keywords

- Representations of algebraic groups
- reductive algebraic groups
- conjugacy classes
- nonabelian 1-cohomology

### Cite this

**On a question of Külshammer for homomorphisms of algebraic groups.** / Lond, Daniel; Martin, Benjamin.

Research output: Contribution to journal › Article

*Journal of Algebra*, vol. 497, pp. 164-198. https://doi.org/10.1016/j.jalgebra.2017.08.031

}

TY - JOUR

T1 - On a question of Külshammer for homomorphisms of algebraic groups

AU - Lond, Daniel

AU - Martin, Benjamin

N1 - Some of the work in this paper was carried out by the first author during his PhD [15]. Both authors acknowledge the financial support of Marsden Grants UOC0501, UOC1009 and UOA1021. We are grateful to Dave Benson and Günter Steinke for helpful conversations. We also thank the referee for their careful reading of the paper.

PY - 2018/3/1

Y1 - 2018/3/1

N2 - Let G be a linear algebraic group over an algebraically closed field of characteristic p≥0. We show that if H1 and H2 are connected subgroups of G such that H1 and H2 have a common maximal unipotent subgroup and H1/Ru(H1) and H2/Ru(H2) are semisimple, then H1 and H2 are G-conjugate. Moreover, we show that if H is a semisimple linear algebraic group with maximal unipotent subgroup U then for any algebraic group homomorphism σ:U→G, there are only finitely many G-conjugacy classes of algebraic group homomorphisms ρ:H→G such that ρ|U is G-conjugate to σ. This answers an analogue for connected algebraic groups of a question of B. Külshammer. In Külshammer's original question, H is replaced by a finite group and U by a Sylow p-subgroup of H; the answer is then known to be no in general. We obtain some results in the general case when H is non-connected and has positive dimension. Along the way, we prove existence and conjugacy results for maximal unipotent subgroups of non-connected linear algebraic groups. When G is reductive, we formulate Külshammer 's question and related conjugacy problems in terms of the nonabelian 1-cohomology of unipotent radicals of parabolic subgroups of G, and we give some applications of this cohomological approach. In particular, we analyse the case when G is a semisimple group of rank 2.

AB - Let G be a linear algebraic group over an algebraically closed field of characteristic p≥0. We show that if H1 and H2 are connected subgroups of G such that H1 and H2 have a common maximal unipotent subgroup and H1/Ru(H1) and H2/Ru(H2) are semisimple, then H1 and H2 are G-conjugate. Moreover, we show that if H is a semisimple linear algebraic group with maximal unipotent subgroup U then for any algebraic group homomorphism σ:U→G, there are only finitely many G-conjugacy classes of algebraic group homomorphisms ρ:H→G such that ρ|U is G-conjugate to σ. This answers an analogue for connected algebraic groups of a question of B. Külshammer. In Külshammer's original question, H is replaced by a finite group and U by a Sylow p-subgroup of H; the answer is then known to be no in general. We obtain some results in the general case when H is non-connected and has positive dimension. Along the way, we prove existence and conjugacy results for maximal unipotent subgroups of non-connected linear algebraic groups. When G is reductive, we formulate Külshammer 's question and related conjugacy problems in terms of the nonabelian 1-cohomology of unipotent radicals of parabolic subgroups of G, and we give some applications of this cohomological approach. In particular, we analyse the case when G is a semisimple group of rank 2.

KW - Representations of algebraic groups

KW - reductive algebraic groups

KW - conjugacy classes

KW - nonabelian 1-cohomology

U2 - 10.1016/j.jalgebra.2017.08.031

DO - 10.1016/j.jalgebra.2017.08.031

M3 - Article

VL - 497

SP - 164

EP - 198

JO - Journal of Algebra

JF - Journal of Algebra

SN - 0021-8693

ER -