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July 2018 A complete description of the antipodal set of most symmetric spaces of compact type
Jonas Beyrer
Osaka J. Math. 55(3): 567-586 (July 2018).

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 $r\neq q>0$.

Citation

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Jonas Beyrer. "A complete description of the antipodal set of most symmetric spaces of compact type." Osaka J. Math. 55 (3) 567 - 586, July 2018.

Information

Published: July 2018
First available in Project Euclid: 4 July 2018

zbMATH: 06927828
MathSciNet: MR3824847

Subjects:
Primary: 22E46, 53C30

Rights: Copyright © 2018 Osaka University and Osaka City University, Departments of Mathematics

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Vol.55 • No. 3 • July 2018
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