Pseudo-Riemannian symmetric spaces: uniform realizations, and open embeddings into Grassmanians

Yu. A. Neretin

Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part III, Tome 256 (1999), pp. 145-167

Résumé

This paper aims on two major observations. The first is that all 54 series of classical symmetric spaces admit simple uniform realizations. Namely, a point of a symmetric space is represented by a pair of complementary linear subspaces $V_1$, $V_2$ in $\mathbb R^k$, $\mathbb C^k$ or $\mathbb H^k$, subject to simple conditions (subspaces may be isotropic, or orthogonal, or rigged with an operator permuting $V_1$ and $V_2$). This observation allows one to work with arbitrary classical symmetric spaces by applying simple elementary methods. The second observation is that there always exist an open embedding of a classical symmetric space into a Grassmanian.