Nilradicals of parabolic subalgebras admitting symplectic structures

TL;DR

This paper classifies all nilradicals of parabolic subalgebras in split real simple Lie algebras that admit symplectic structures, using Kostant's highest weight vectors and cohomology conditions.

Contribution

It provides a complete list of symplectic nilradicals in parabolic subalgebras of split real simple Lie algebras, employing cohomological and representation-theoretic methods.

Findings

Identifies all nilradicals with symplectic structures

Uses Kostant's description of highest weight vectors

Derives necessary conditions for symplectic structures

Abstract

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.