On a question of Külshammer for homomorphisms of algebraic groups

Daniel Lond, Benjamin Martin

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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
Early online date2 Oct 2017
Publication statusPublished - 1 Mar 2018

Bibliographical note

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.


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


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