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 Gconjugate. 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 Gconjugacy classes of algebraic group homomorphisms ρ:H→G such that ρU is Gconjugate 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 psubgroup of H; the answer is then known to be no in general. We obtain some results in the general case when H is nonconnected and has positive dimension. Along the way, we prove existence and conjugacy results for maximal unipotent subgroups of nonconnected linear algebraic groups. When G is reductive, we formulate Külshammer 's question and related conjugacy problems in terms of the nonabelian 1cohomology 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.
In Külshammer's original question, H is replaced by a finite group and U by a Sylow psubgroup of H; the answer is then known to be no in general. We obtain some results in the general case when H is nonconnected and has positive dimension. Along the way, we prove existence and conjugacy results for maximal unipotent subgroups of nonconnected linear algebraic groups. When G is reductive, we formulate Külshammer 's question and related conjugacy problems in terms of the nonabelian 1cohomology 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 

Pages (fromto)  164198 
Number of pages  35 
Journal  Journal of Algebra 
Volume  497 
Early online date  2 Oct 2017 
DOIs  
Publication status  Published  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.Keywords
 Representations of algebraic groups
 reductive algebraic groups
 conjugacy classes
 nonabelian 1cohomology
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Ben Martin
 School of Natural & Computing Sciences, Mathematical Science  Personal Chair
Person: Academic