Square-integrable representations and the coadjoint action of solvable Lie groups

TL;DR

This paper characterizes square-integrable representations of solvable Lie groups via coadjoint orbits, establishing a bijection with open orbits and exploring the structure of primitive ideals.

Contribution

It provides a new characterization of square-integrable representations in terms of coadjoint orbits and analyzes the structure of primitive ideals for solvable Lie groups.

Findings

Square-integrable representations correspond to simply connected open coadjoint orbits.

Open coadjoint orbits imply the existence of a compact open subset in the primitive ideal space.

Isolated points of the primitive ideal space are of type I when the nilradical has codimension 1.

Abstract

We characterize the square-integrable representations of (connected, simply connected) solvable Lie groups in terms of the generalized orbits of the coadjoint action. We prove that the normal representations corresponding, via the Puk\'anszky correspondence, to open coadjoint orbits are type I, not necessarily square-integrable representations. We show that the quasi-equivalence classes of type I square-integrable representations are in bijection with the simply connected open coadjoint orbits, and the existence of an open coadjoint orbit guarantees the existence of a compact open subset of the space of primitive ideals of the group. When the nilradical has codimension 1, we prove that the isolated points of the primitive ideal space are always of type I. This is not always true for codimension greater than 2, as shown by specific examples of solvable Lie groups that have dense, but not locally closed, coadjoint orbits.