On involutions in the Weyl group and $B$-orbit closures in the orthogonal case

TL;DR

This paper investigates the structure and closure relations of Borel group orbits in the dual of the Lie algebra of the unipotent radical in complex orthogonal groups, linking orbit closures to Bruhat order on basis involutions.

Contribution

It establishes a Bruhat order criterion for orbit closure containment and computes orbit dimensions, providing new insights into orbit structure in the orthogonal case.

Findings

Orbit closure containment corresponds to Bruhat order.

Dimensions of orbits are explicitly computed.

Conjectural description of orbit closures is proposed.

Abstract

We study coadjoint $B$-orbits on $\mathfrak{n}^*$, where $B$ is a Borel subgroup of a complex orthogonal group $G$, and $\mathfrak{n}$ 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 $\Omega_w$. We prove that, given basis involutions $\sigma$, $\tau$ in $W$, if the orbit $\Omega_{\sigma}$ is contained in the closure of the orbit $\Omega_{\tau}$ then $\sigma$ is less than or equal to $\tau$ with respect to the Bruhat order on $W$. For a basis involution $w$, we also compute the dimension of $\Omega_w$ and present a conjectural description of the closure of $\Omega_w$.