Square-integrable representations and the coadjoint action of solvable Lie groups
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Ingrid Beltiţă
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.
Funding source: Ministerul Cercetării, Inovării şi Digitalizării
Award Identifier / Grant number: PN-III-P4-ID-PCE-2020-0878
Funding statement: The research of the second-named author was supported by a grant of the Ministry of Research, Innovation and Digitization, CNCS/CCCDI – UEFISCDI, project number PN-III-P4-ID-PCE-2020-0878, within PNCDI III.
Acknowledgements
We are grateful to Karl-Hermann Neeb for useful remarks on 1-connected Lie groups, and to Jordy van Velthoven for pointing out an inaccuracy in an earlier version of our paper, concerning the centre of the group 𝐺 in Example 7.1. We also thank the referee for suggestions that improved our paper at many places.
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Communicated by: Jan Bruinier
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Articles in the same Issue
- Frontmatter
- Square-integrable representations and the coadjoint action of solvable Lie groups
- Deviation probabilities for extremal eigenvalues of large Chiral non-Hermitian random matrices
- Nef vector bundles on a hyperquadric with first Chern class two
- Scaling spectrum of a class of self-similar measures with product form on ℝ
- Idempotent decomposition of regularity and characterization for the accumulation of associated spectrum
- Geodesic orbit Randers metrics in homogeneous bundles over generalized Stiefel manifolds
- Proof of some conjectures of Guo and of Tang
- On Absolute and Quantitative Subspace Theorems
- Birational equivalence of the Zassenhaus varieties for basic classical Lie superalgebras and their purely-even reductive Lie subalgebras in odd characteristic
- Geometric approach to the Moore–Penrose inverse and the polar decomposition of perturbations by operator ideals
- The Weil bound for generalized Kloosterman sums of half-integral weight
- Relative Rota–Baxter groups and skew left braces
- Asai gamma factors over finite fields
- Representations of non-finitely graded Lie algebras related to Virasoro algebra
- Multiple solutions for fractional elliptic systems
- Arithmetic Bohr radius for the Minkowski space
- On Strichartz estimates for many-body Schrödinger equation in the periodic setting
Articles in the same Issue
- Frontmatter
- Square-integrable representations and the coadjoint action of solvable Lie groups
- Deviation probabilities for extremal eigenvalues of large Chiral non-Hermitian random matrices
- Nef vector bundles on a hyperquadric with first Chern class two
- Scaling spectrum of a class of self-similar measures with product form on ℝ
- Idempotent decomposition of regularity and characterization for the accumulation of associated spectrum
- Geodesic orbit Randers metrics in homogeneous bundles over generalized Stiefel manifolds
- Proof of some conjectures of Guo and of Tang
- On Absolute and Quantitative Subspace Theorems
- Birational equivalence of the Zassenhaus varieties for basic classical Lie superalgebras and their purely-even reductive Lie subalgebras in odd characteristic
- Geometric approach to the Moore–Penrose inverse and the polar decomposition of perturbations by operator ideals
- The Weil bound for generalized Kloosterman sums of half-integral weight
- Relative Rota–Baxter groups and skew left braces
- Asai gamma factors over finite fields
- Representations of non-finitely graded Lie algebras related to Virasoro algebra
- Multiple solutions for fractional elliptic systems
- Arithmetic Bohr radius for the Minkowski space
- On Strichartz estimates for many-body Schrödinger equation in the periodic setting
