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

TL;DR

This paper investigates conjugacy and homomorphism classification of algebraic groups, extending K"ulshammer's question from finite groups to connected algebraic groups, with applications to cohomology and specific cases like rank 2 semisimple groups.

Contribution

It provides new conjugacy results for connected algebraic subgroups sharing unipotent parts and classifies homomorphisms from maximal unipotent subgroups, extending K"ulshammer's question.

Findings

Connected subgroups with common unipotent parts are conjugate in G.

Finiteness of conjugacy classes of homomorphisms from semisimple groups.

Cohomological framework for conjugacy problems in reductive groups.

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.