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

Abstract

Let $G$ be the symplectic group, $\Phi =C_n$ its root system, $B\subset G$ its standard Borel subgroup, $W$ the Weyl group of $\Phi$. To each involution $\sigma \in W$ one can assign the $B$-orbit ${\Omega }_{\sigma }$ contained in the dual space of the Lie algebra of the unipotent radical of $B$. We prove that ${\Omega }_{\sigma }$ is contained in the Zariski closure of ${\Omega }_{\tau }$ if and only of $\sigma \le \tau$ with respect to the Bruhat--Chevalley order. We also prove that $\dim {\Omega }_{\sigma }$ is equal to $l(\sigma )$, the length of $\sigma$ in $W$.

arXiv comment13 pages