# Minimal Parabolic $k$-subgroups acting on Symmetric $k$-varieties Corresponding to $k$-split Groups

## TL;DR

This paper investigates how minimal parabolic $k$-subgroups act on symmetric $k$-varieties, introducing a map that relates their orbits to those over algebraically closed fields, with conditions for surjectivity based on group structure.

## Contribution

It defines a new map linking orbits of minimal parabolic $k$-subgroups on symmetric $k$-varieties to algebraically closed field orbits and establishes surjectivity conditions for $k$-split groups.

## Key findings

- A map embedding orbits is constructed.
- Surjectivity depends on the dimension of a maximal $k$-split torus.
- Conditions are established specifically for $k$-split groups.

## Abstract

Symmetric $k$-varieties are a natural generalization of symmetric spaces to general fields $k$. We study the action of minimal parabolic $k$-subgroups on symmetric $k$-varieties and define a map that embeds these orbits within the orbits corresponding to algebraically closed fields. We develop a condition for the surjectivity of this map in the case of $k$-split groups that depends only on the dimension of a maximal $k$-split torus contained within the fixed point group of the involution defining the symmetric $k$-variety.