The Bruhat order on Hermitian symmetric varieties and on abelian nilradicals

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 p^u be the Lie algebra of P^u and L a Levi factor of P. Then L is a Hermitian symmetric subgroup of G and B acts with finitely many orbits both on p^u and on G/L. In this paper we study the Bruhat order of the B-orbits in p^u and in G/L, proving respectively a conjecture of Panyushev and a conjecture of Richardson and Ryan.