The Bruhat order on Hermitian symmetric varieties and on abelian nilradicals

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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 L 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.