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 Pu, and let B be a Borel subgroup of G contained in P. Let pu be the Lie algebra of Pu 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 pu and on G/L. In this paper we study the Bruhat order of the B-orbits in pu and in G/L, proving respectively a conjecture of Panyushev and a conjecture of Richardson and Ryan.