Lipsman mapping and dual topology of semidirect products

Rahali, Aymen

doi:10.36045/bbms/1553047234

Cited by 5 publications

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“…Let π and ρ be unitary representations of G. It is clear that if π is contained in ρ (we write π ρ), then π ≺ ρ. Relativized to G, the relationship of weak containment gives the operation of closure in the hull-kernel topology (the dual topology). A lot of the problems in harmonic analysis on G are based on explicit determination of the dual topology of the group G (for example, see [4], [14], [29], [30], [33]).…”

Section: Introductionmentioning

confidence: 99%

Kirillov-Lipsman orbit method of a class of Gelfand pairs

Rahali

2023

Preprint

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Let $G:=K\ltimes N$ be the semidirect product with Lie algebra $\mathfrak{g},$ where $N$ is a connected and simply connected nilpotent Lie group, and $K$ is a subgroup of the automorphism group, $Aut(N),$ of $N.$ We say that the pair $(K,N)$ is a nilpotent Gelfand pair when the set $L_K^1(N)$ of integrable $K$-invariant functions on $N$ forms an abelian algebra under convolution.According to Lipsman, the unitary dual $\widehat{G}$ of $G$ is in one-to-one correspondence with the space of admissible coadjoint orbits $\mathfrak{g}^\ddag/G$ of $G.$ Under some assumptions on the pair $(K,N)$ we will show in this work, that the quotient topology of $\mathfrak{g}^\ddag/G$ could be read from the usual topology of some parameters space $\mathcal{P},$ called Mackey's parameters space (will be defined below in section 3). Furthermore, we show that the Kirillov-Lipsman bijection $$\widehat{G}\simeq\mathfrak{g}^\ddagger/G$$is a homeomorphism for a class of Lie groups associated with the nilpotent Gelfand pairs $(K,N).$

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Section: Introductionmentioning

confidence: 99%

Kirillov-Lipsman orbit method of a class of Gelfand pairs

Rahali

2023

Preprint

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Orbit method and dual topology for certain Lie groups

Rahali

Chenni

Selmi

2022

Banach J. Math. Anal.

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Branching rule and coadjoint orbit for Heisenberg Gelfand pairs

Rahali¹

2023

Int. J. Math.

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Let [Formula: see text] be a closed connected subgroup of the unitary group [Formula: see text] and let [Formula: see text] be the [Formula: see text]-dimensional Heisenberg group ([Formula: see text]). We consider the semidirect product [Formula: see text] such that [Formula: see text] is a Gelfand pair. Let [Formula: see text] be the respective Lie algebras of [Formula: see text] and [Formula: see text] and [Formula: see text] be the natural projection. It was pointed out by Lipsman, that the unitary dual [Formula: see text] of [Formula: see text] is in one-to-one correspondence with the space of admissible coadjoint orbits [Formula: see text] (see [10]). Let [Formula: see text] be a generic representation of [Formula: see text] and let [Formula: see text] To these representations we associate, respectively, the admissible coadjoint orbit [Formula: see text] and [Formula: see text] (via the Lipsman’s correspondence). We denote by [Formula: see text] the number of [Formula: see text]-orbits in [Formula: see text] which is called the Corwin–Greenleaf multiplicity function. The Kirillov–Lipsman’s orbit method suggests that the multiplicity [Formula: see text] of an irreducible [Formula: see text]-module [Formula: see text] occurring in the restriction [Formula: see text] could be read from the coadjoint action of [Formula: see text] on [Formula: see text] In this paper, we show that [Formula: see text] For the special case [Formula: see text] (maximal torus in the unitary group [Formula: see text]) and [Formula: see text], we prove that [Formula: see text]

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Lipsman mapping and dual topology of semidirect products

Cited by 5 publications

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Kirillov-Lipsman orbit method of a class of Gelfand pairs

Kirillov-Lipsman orbit method of a class of Gelfand pairs

Orbit method and dual topology for certain Lie groups

Branching rule and coadjoint orbit for Heisenberg Gelfand pairs

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