On Involutions in the Weyl Group and B-Orbit Closures in the Orthogonal Case

Abstract

We study coadjoint B-orbits on n * , where B is a Borel subgroup of a complex orthogonal group G, and n is the Lie algebra of the unipotent radical of B. To each basis involution w in the Weyl group W of G one can assign the associated B-orbit Ω w . We prove that, given basis involutions σ, τ in W , if the orbit Ω σ is contained in the closure of the orbit Ω τ then σ is less than or equal to τ with respect to the Bruhat order on W . For a basis involution w, we also compute the dimension of Ω w and present a conjectural description of the closure of Ω w .

Nilpotent orbits of height 2 and involutions in the affine Weyl group

“…Finally, we mention the work of Ignatyev [14], [15], [16]. If G is a classical group, in these papers the author attach to any involution in W a coadjoint B-orbit in the dual Lie algebra u * (where u denotes the nilradical of b), and the partial order J o u r n a l P r e -p r o o f among these orbits is studied in terms of the Bruhat order of the corresponding involutions.…”