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

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 $Spin(2r+1)r\ge 5$, $E_8$ 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 $G{r}_{r,r+q}(\mathbb{R})$ has a $rq$-dimensional orbit as antipodal set if $r\ge 5$ and $q>0$.

arXiv comment25 pages; V2 includes corrections of some calculations