# Conjugacy of Levi subgroups of reductive groups and a generalization to linear algebraic groups

## TL;DR

This paper studies the conjugacy properties of Levi subgroups in reductive algebraic groups and extends the results to general linear algebraic groups using pseudo-parabolic subgroups.

## Contribution

It provides a parametrization of Levi subgroup conjugacy classes and establishes conditions for rational and geometric conjugacy, generalizing to linear algebraic groups.

## Key findings

- Levi K-subgroups are rationally conjugate iff they are geometrically conjugate.
- Parametrization of conjugacy classes via simple roots.
- Extension of results to linear algebraic groups using pseudo-parabolic subgroups.

## Abstract

We investigate Levi subgroups of a connected reductive algebraic group G, over a ground field K. We parametrize their conjugacy classes in terms of sets of simple roots and we prove that two Levi K-subgroups of G are rationally conjugate if and only if they are geometrically conjugate. These results are generalized to arbitrary connected linear algebraic K-groups. In that setting the appropriate analogue of a Levi subgroup is derived from the notion of a pseudo-parabolic subgroup.