Nilradicals of parabolic subalgebras admitting symplectic structures

In this paper we describe all the nilradicals of parabolic subalgebras of split real simple Lie algebras admitting symplectic structures. The main tools used to obtain this list are Kostant’s description of the highest weight vectors (hwv) of the cohomology of these nilradicals and some necessary conditions obtained for the $\mathfrak{g}$-hwv’s of $H^2(\mathfrak{n})$ for a finite dimensional real symplectic nilpotent Lie algebra $\mathfrak{n}$ with a reductive Lie subalgebra of derivations $\mathfrak{g}$ acting on it.