The Bruhat--Chevalley order on involutions of the hyperoctahedral group and~combinatorics of $B$-orbit closures

M. V. Ignat'ev

Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 23, Tome 400 (2012), pp. 166-188

Résumé

Let $G=\mathrm{Sp}_{2n}(\mathbb C)$ be the symplectic group, $B$ its Borel subgroup and $\Phi=C_n$ the root system of $G$. To each involution $\sigma$ in the Weyl group $W$ of $\Phi$ one can assign the orbit $\Omega_\sigma$ of the coadjoint action of $B$ on the dual space of the Lie algebra of the unipotent radical of $B$. Let $\sigma,\tau$ be involutions in $W$. We prove that $\Omega_\sigma$ is contained in the closure of $\Omega_\tau$ if and only if $\sigma$ is less or equal than $\tau$ with respect to the Bruhat–Chevalley order on $W$.