Multiplicity-free representations and coisotropic actions of certain nilpotent Lie groups over quasi-symmetric Siegel domains

We study multiplicity-free representations of nilpotent Lie groups over a quasi-symmetric Siegel domain, with a focus on certain two-step nilpotent Lie groups. We provide necessary and sufficient conditions for the multiplicity-freeness property. Specifically, we establish the equivalence between the disjointness of irreducible unitary representations realized over the domain, the multiplicity-freeness of the unitary representation on the space of $L^2$ holomorphic functions, and the coisotropicity of the group action.