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, Algèbre). For a pair of totally isotropic subspaces u, v ≤ V, the double cosets P_uγP_v of their stabilizers P_u, P_v in Γ = SU(V ) are considered. A description of the cosets P_uγP_v in terms of the intersection distance d_in(u, γ(v)) and the Witt index of u + γ(v) is given.