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.