The Bruhat order on Hermitian symmetric varieties and on abelian nilradicals

TL;DR

This paper investigates the structure of the Bruhat order on Borel orbits in certain algebraic varieties related to Hermitian symmetric spaces, confirming two conjectures about their ordering.

Contribution

It proves two conjectures regarding the Bruhat order on Borel orbits in abelian nilradicals and Hermitian symmetric varieties.

Findings

Confirmed Panyushev's conjecture on Bruhat order in abelian nilradicals.

Confirmed Richardson and Ryan's conjecture on Bruhat order in Hermitian symmetric varieties.

Established structural properties of Borel orbits in these algebraic contexts.

Abstract

Let $G$ be a simple algebraic group and $P$ a parabolic subgroup of $G$ with abelian unipotent radical $P^u$, and let $B$ be a Borel subgroup of $G$ contained in P. Let $\mathfrak{p}^u$ be the Lie algebra of $P^u$ and let $L$ be a Levi factor of $P$, then $L$ is a Hermitian symmetric subgroup of $G$ and $B$ acts with finitely many orbits both on $\mathfrak{p}^u$ and on $G/L$. In this paper we study the Bruhat order of the $B$-orbits in $\mathfrak{p}^u$ and in $G/L$, proving respectively a conjecture of Panyushev and a conjecture of Richardson and Ryan.