Double Cosets of Stabilizers of Totally Isotropic Subspaces in a Special Unitary Group. I

Abstract

Let D be a division algebra with a fixed involution, and let V be the corresponding unitary space over D with T -condition (see N. Bourbaki, Algebre). For a pair of totally isotropic subspaces u, v ≤ V, the double cosets P u γP v of their stabilizers P u , P v in Γ = SU(V ) are considered. A description of the cosets P u γP v in terms of the intersection distance din(u, γ(v)) and the Witt index of u + γ(v) is given.