The Bruhat order on Hermitian symmetric varieties and on abelian nilradicals

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.