# Complexity $c$ Pairs in Simple Algebraic Groups

## TL;DR

This paper classifies pairs of subgroups in simple algebraic groups based on the complexity of their multiplication action, focusing on cases where the complexity is 0 or 1, revealing new classifications related to horospherical spaces.

## Contribution

It provides a classification of complexity 0 and 1 subgroup pairs in simple algebraic groups, especially when only one subgroup is reductive, involving horospherical homogeneous spaces.

## Key findings

- Reductive subgroups cannot form complexity 0 or 1 pairs with each other.
- Classification of pairs with exactly one reductive subgroup involves small rank horospherical spaces.
- Results include classification of diagonal spherical actions on products of flag varieties and affine homogeneous spaces.

## Abstract

We call a pair of closed subgroups $(G_1,G_2)$ from a connected reductive algebraic group $G$ a {\it complexity $c$ pair} if the multiplication action of the pair on $G$ is of complexity $c$. The main focus of this article is on the cases where $G$ is simple and $c$ is either 0 or 1. After showing that both of the subgroups $G_1$ and $G_2$ cannot be reductive subgroups unless $c>1$, we look for the cases where exactly one of the subgroups $G_1$ and $G_2$ is reductive. It turns out that there are only a few such pairs, and their classification involves the horospherical homogeneous spaces of small ranks. As a byproduct of the circle of ideas that we use for this development, we obtain the classification of the diagonal spherical actions of simple algebraic groups on the products of flag varieties with affine homogeneous spaces.