Coadjoint Orbits of Reductive Type of Parabolic and Seaweed Lie Subalgebras

A connected algebraic group Q defined over a field of characteristic zero is quasi-reductive if there is an element of $\mathfrak q^*$ of reductive type, that is such that the quotient of its stabilizer by the center of Q is a reductive subgroup of ${\mathrm GL}(\mathfrak q)$. Such groups appear in harmonic analysis when unitary representations are studied. In particular, over the field of real numbers they turn out to be the groups with discrete series and their irreducible unitary square integrable representations are parameterised by coadjoint orbits of reductive type. Due to the results of Duflo, coadjoint representation of a quasi-reductive Q possesses a so-called maximal reductive stabiliser and knowing this subgroup, defined up to a conjugation in Q, one can describe all coadjoint orbits of reductive type. In this paper, we consider quasi-reductive parabolic subalgebras of simple complex Lie algebras as well as seaweed subalgebras of $\mathfrak {gl}_n(\mathbb C)$ and describe the classes of their maximal reductive stabilizers.