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

Ingrid Beltita, Daniel Beltita

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 Pukánszky 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.

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

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 Pukánszky 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.

Paper Structure

This paper contains 11 sections, 24 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on generalized orbits of the coadjoint action
  3. Pukánszky theory for solvable Lie groups
  4. Open coadjoint quasi-orbits
  5. Square integrable representations and the corresponding generalized coadjoint orbits
  6. Unimodular solvable Lie groups do not admit square-integrable representations
  7. Open orbits, type I representations, and compact open subsets of the primitive ideal space
  8. Open quasi-orbits and type ${\rm I}$ representations
  9. Open coadjoint orbits and compact open subsets of ${\rm Prim}(G)$
  10. Solvable Lie groups with nilradicals of codimension $1$
  11. Examples

Key Result

Lemma \oldthetheorem

For every $\xi \in {\mathfrak g}^*$, $g \in G(\xi)$, and $\chi \in \mathop{G}\limits^{{{$∧$} \space }}(\xi)$ we have

Theorems & Definitions (55)

  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • proof
  • Remark \oldthetheorem
  • Proposition \oldthetheorem
  • proof
  • Corollary \oldthetheorem
  • proof
  • Proposition \oldthetheorem
  • ...and 45 more