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

TL;DR

This paper investigates when certain nilpotent Lie groups acting on quasi-symmetric Siegel domains have multiplicity-free representations, establishing conditions linking representation disjointness, Bergman space properties, and coisotropic actions.

Contribution

It provides necessary and sufficient conditions for multiplicity-freeness of representations of specific nilpotent Lie groups on these domains, connecting geometric and representation-theoretic properties.

Findings

Disjointness of irreducible unitary representations is equivalent to multiplicity-freeness.

Multiplicity-freeness is equivalent to the coisotropicity of the group action.

Established a characterization linking geometric action properties with representation multiplicity.

Abstract

We study multiplicity-free representations of 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 to hold. 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 Bergman space, and the coisotropicity of the group action.