Exceptional representations of a double quiver of type A, and Richardson elements in seaweed Lie algebras

In this paper, we study the set of $\mathrm{\Delta }$-filtered modules of quasi-hereditary algebras arising from quotients of the double of quivers of type $A$. Our main result is that for any fixed $\mathrm{\Delta }$-dimension vector, there is a unique (up to isomorphism) exceptional $\mathrm{\Delta }$-filtered module. We then apply this result to show that there is always an open adjoint orbit in the nilpotent radical of a seaweed Lie algebra in ${\mathrm{g}\mathrm{l}}_n(\text{field})$, thus answering positively in this ${\mathrm{g}\mathrm{l}}_n(\text{field})$ case to a question raised independently by Michel Duflo and Dmitri Panyushev. An example of a seaweed Lie algebra in a simple Lie algebra of type $E_8$ not admitting an open orbit in its nilpotent radical is given.