On a question of Külshammer for homomorphisms of algebraic groups

Daniel Lond, Benjamin Martin

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Abstract

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.
Original languageEnglish
Pages (from-to)164-198
Number of pages35
JournalJournal of Algebra
Volume497
Early online date2 Oct 2017
DOIs
Publication statusPublished - 1 Mar 2018

Fingerprint

Linear Algebraic Groups
Maximal Subgroup
Algebraic Groups
Homomorphisms
Semisimple
Subgroup
Conjugacy Problem
Semisimple Groups
Parabolic Subgroup
Conjugacy
Conjugacy class
Algebraically closed
Homomorphism
Cohomology
Finite Group
Analogue

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.

In: Journal of Algebra, Vol. 497, 01.03.2018, p. 164-198.

Research output: Contribution to journalArticle

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abstract = "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. K{\"u}lshammer. In K{\"u}lshammer'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 K{\"u}lshammer '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.",
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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.

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