# On a question of K\"ulshammer for homomorphisms of algebraic groups

Research paper by **Daniel Lond, Benjamin Martin**

Indexed on: **30 Oct '16**Published on: **30 Oct '16**Published in: **arXiv - Mathematics - Group Theory**

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#### Abstract

Let $G$ be a linear algebraic group over an algebraically closed field of
characteristic $p\geq 0$. We show that if $H_1$ and $H_2$ are connected
subgroups of $G$ such that $H_1$ and $H_2$ have a common maximal unipotent
subgroup and $H_1/R_u(H_1)$ and $H_2/R_u(H_2)$ are semisimple, then $H_1$ and
$H_2$ 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 $\sigma\colon U\rightarrow G$, there are only finitely many
$G$-conjugacy classes of algebraic group homomorphisms $\rho\colon H\rightarrow
G$ such that $\rho|_U$ is $G$-conjugate to $\sigma$. This answers an analogue
for connected algebraic groups of a question of B. K\"ulshammer.
In K\"ulshammer'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\"ulshammer'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.