Symmetric spaces as Grassmannians

Abstract

In the PhD thesis of Huang with Leung (Huang, A uniform description of Riemannian symmetric spaces as Grassmannians using magic square, www.ims.cuhk.edu.hk/~leung/; Huang and Leung, Math Ann 350:76–106, 2010), all compact symmetric spaces are represented as (structured) Grassmannians over the algebra ${\mathbb{K L}:=\mathbb{K} \otimes_{\mathbb{R}} \mathbb{L}}$ where ${\mathbb{K},\mathbb{L}}$ are real division algebras. This was known in some (infinitesimal) sense for exceptional spaces (see Baez, Bull Am Math Soc 39:145–205, 2001); the main purpose in Huang (www.ims.cuhk.edu.hk/~leung/) and Huang and Leung (Math Ann 350:76–106, 2010) was to give a similar description for the classical spaces. In the present paper we give a different approach to this result by investigating the fixed algebras ${\mathbb{B}}$ of involutions on ${\mathbb{A} =\mathbb{K}\mathbb{L}}$ with half-dimensional eigenspaces together with the automorphism groups of ${\mathbb{A}}$ and ${\mathbb{B}}$. We also relate the results to the classification of self-reflective submanifolds in Chen and Nagano (Trans Am Math Soc 308:273–297, 1988) and Leung (J Differ Geom 14:167–177, 1979).