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

Abstract

In 2016, the authors considered the decomposition \( \mathrm{SU}\left(D,h\right)=\underset{i}{\cup }{P}_u{\gamma}_i{P}_{\upsilon } \), where SU(D, h) is a special unitary group over a division algebra D with involution, h is a symmetric or skew-symmetric nondegenerate Hermitian form, and P_u, P_υ are stabilizers of totally isotropic subspaces of the unitary space. Since Γ = SU(D, h) is a point group of a classical algebraic group \( \tilde{\Gamma} \), there is the “order of adherence” on the set of double cosets {P_uγ_iP_υ}, which is induced by the Zariski topology on \( \tilde{\Gamma} \). In the present paper, the adherence of such double cosets is described for the cases where \( \tilde{\Gamma} \) is an orthogonal or a symplectic group (that is, for groups of types B_r, C_r, D_r).