On a question of Külshammer for homomorphisms of algebraic groups

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.