Pseudoriemannian symmetric spaces: one-type realizations and open embeddings to grassmannians

Abstract

We show that each classical pseudoriemann symmetric space G/H can be realized as space of pairs of complementary subspaces in a linear space. For each classical symmetric space we construct an open embedding to a grassmannian or to a product of two grassmanianns. We also show that the representation of the group G in L^2 on G/H is equivalent to restriction of a degenerated principal series representation of some group Q containing G.