Osaka Journal of Mathematics

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

Jonas Beyrer

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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 $r\neq q>0$.

Article information

Source
Osaka J. Math., Volume 55, Number 3 (2018), 567-586.

Dates
First available in Project Euclid: 4 July 2018

Permanent link to this document
https://projecteuclid.org/euclid.ojm/1530691244

Mathematical Reviews number (MathSciNet)
MR3824847

Zentralblatt MATH identifier
06927828

Subjects
Primary: 22E46: Semisimple Lie groups and their representations 53C30: Homogeneous manifolds [See also 14M15, 14M17, 32M10, 57T15]

Citation

Beyrer, Jonas. A complete description of the antipodal set of most symmetric spaces of compact type. Osaka J. Math. 55 (2018), no. 3, 567--586. https://projecteuclid.org/euclid.ojm/1530691244


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