In this paper, we show that there is always an open adjoint orbit in the nilpotent radical of a seaweed Lie algebra in gln(k), thus answering positively in this gln(k) case to a question raised independently by Michel Duflo and Dmitri Panyushev. The proof gives an explicit construction, using �-filtered modules of quasi-hereditary algebras arising from quotients of the double of quivers of type A. An example of a seaweed Lie algebra in a simple Lie algebra of type E8 not admitting an open orbit in its nilpotent radical is given.
Jensen, B. T., Su, X., & Yu, R. W. T. (2009). Rigid representations of a double quiver of type A, and Richardson elements in seaweed Lie algebras. Bulletin of the London Mathematical Society, 41(1), 1-15. https://doi.org/10.1112/blms/bdn087