A complete description of the antipodal set of most symmetric spaces of compact type

TL;DR

This paper characterizes the dimensions of antipodal sets in most symmetric spaces of compact type, revealing their structure as unions of specific $K$-orbits and providing explicit dimensions for various classical and exceptional cases.

Contribution

It provides a complete description of the antipodal sets in most irreducible symmetric spaces of compact type, detailing their orbit structures and dimensions, excluding certain spaces with specific root systems and fundamental groups.

Findings

Antipodal sets in certain Lie groups are single orbits with specific dimensions.

Explicit orbit dimensions are given for classical and exceptional symmetric spaces.

The structure of antipodal sets varies depending on the space's root system and fundamental group.

Abstract

It is known that the antipodal set of a Riemannian symmetric space of compact type $G / K$ consists of a union of $K$-orbits. We determine the dimensions of these $K$-orbits of most irreducible symmetric spaces of compact type. The symmetric spaces we are not going to deal with are those with restricted root system $\mathfrak{a}_r$ and a non-trivial fundamental group, which is not isomorphic to $\mathbb{Z}_2$ or $\mathbb{Z}_{r+1}$. For example, we show that the antipodal sets of the Lie groups $Spin(2r+1)\:\: r\geq 5$, $E_8$ and $G_2$ consist only of one orbit which is of dimension $2r$, 128 and 6, respectively; $SO(2r+1)$ has also an antipodal set of dimension $2r$; and the Grassmannian $Gr_{r,r+q}(\mathbb{R})$ has a $rq$-dimensional orbit as antipodal set if $r\geq 5$ and $q>0$.