Partial Characters and Signed Quotient Hypergroups

1999 ◽  
Vol 51 (1) ◽  
pp. 96-116 ◽  
Author(s):  
Margit Rösler ◽  
Michael Voit
Keyword(s):  
Heisenberg Group ◽  
Closed Subgroup ◽  
Gelfand Pair ◽  
Coset Space ◽  
Commutative Hypergroup

AbstractIfGis a closed subgroup of a commutative hypergroupK, then the coset spaceK/Gcarries a quotient hypergroup structure. In this paper, we study related convolution structures onK/Gcoming fromdeformations of the quotient hypergroup structure by certain functions onKwhich we call partial characters with respect toG. They are usually not probability-preserving, but lead to so-called signed hypergroups onK/G. A first example is provided by the Laguerre convolution on [0, ∞[, which is interpreted as a signed quotient hypergroup convolution derived from the Heisenberg group. Moreover, signed hypergroups associated with the Gelfand pair (U(n, 1),U(n)) are discussed.

scholarly journals Corwin–Greenleaf multiplicity function for compact extensions of the Heisenberg group

2018 ◽  
Vol 29 (09) ◽  
pp. 1850056
Author(s):  
Majdi Ben Halima ◽  
Anis Messaoud
Keyword(s):  
Heisenberg Group ◽  
Lie Algebras ◽  
Closed Subgroup ◽  
Sufficient Conditions ◽  
Multiplicity Function ◽  
Coadjoint Orbits ◽  
Gelfand Pair ◽  
Irreducible Unitary Representations ◽  
Multiplicity Formula ◽  
The Relationship

Let [Formula: see text] be the [Formula: see text]-dimensional Heisenberg group and [Formula: see text] a closed subgroup of [Formula: see text] acting on [Formula: see text] by automorphisms such that [Formula: see text] is a Gelfand pair. Let [Formula: see text] be the semidirect product of [Formula: see text] and [Formula: see text]. Let [Formula: see text] be the respective Lie algebras of [Formula: see text] and [Formula: see text], and [Formula: see text] the natural projection. For coadjoint orbits [Formula: see text] and [Formula: see text], 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. In this paper, we give two sufficient conditions on [Formula: see text] in order that [Formula: see text] For [Formula: see text], assuming furthermore that [Formula: see text] and [Formula: see text] are admissible and denoting respectively by [Formula: see text] and [Formula: see text] their corresponding irreducible unitary representations, we also discuss the relationship between [Formula: see text] and the multiplicity [Formula: see text] of [Formula: see text] in the restriction of [Formula: see text] to [Formula: see text]. Especially, we study in Theorem 4 the case where [Formula: see text]. This inequality is interesting because we expect the equality as the naming of the Corwin–Greenleaf multiplicity function suggests.

Abstract measure algebras over homogeneous spaces of compact groups

2018 ◽  
Vol 29 (01) ◽  
pp. 1850005 ◽  
Author(s):  
Arash Ghaani Farashahi
Keyword(s):  
Compact Group ◽  
Systematic Study ◽  
Measure Space ◽  
Closed Subgroup ◽  
Homogeneous Spaces ◽  
Compact Groups ◽  
Coset Space ◽  
Left Coset ◽  
Abstract Measure

This paper presents a systematic study for abstract Banach measure algebras over homogeneous spaces of compact groups. Let [Formula: see text] be a closed subgroup of a compact group [Formula: see text] and [Formula: see text] be the left coset space associated to the subgroup [Formula: see text] in [Formula: see text]. Also, let [Formula: see text] be the Banach measure space consists of all complex measures over [Formula: see text]. Then we introduce the abstract notions of convolution and involution over the Banach measure space [Formula: see text].

A Paley-Wiener Theorem for the Spherical Transform Associated with the Generalized Gelfand Pair $(U(p,q),H_{n})$, $p+q=n$

2016 ◽  
Vol 119 (2) ◽  
pp. 249
Author(s):  
Silvina Campos
Keyword(s):  
Heisenberg Group ◽  
Holomorphic Functions ◽  
Real Variable ◽  
Gelfand Pair ◽  
Wiener Theorem ◽  
Spherical Transforms

In this work we prove a Paley-Wiener theorem for the spherical transform associated to the generalized Gelfand pair $(H_n\ltimes U(p,q),H_n)$, where $H_n$ is the $2n+1$-dimensional Heisenberg group. In particular, by using the identification of the spectrum of $(U(p,q),H_n)$ with a subset $\Sigma$ of $\mathbb{R}^2$, we prove that the restrictions of the spherical transforms of functions in $C_{0}^{\infty}(H_n)$ to appropriated subsets of $\Sigma$, can be extended to holomorphic functions on $\mathbb{C}^2$. Also, we obtain a real variable characterizations of such transforms.

scholarly journals On orbits of algebraic groups and Lie groups

1982 ◽  
Vol 25 (1) ◽  
pp. 1-28 ◽  
Author(s):  
R.W. Richardson
Keyword(s):  
Fixed Points ◽  
Homogeneous Space ◽  
Algebraic Group ◽  
Lie Groups ◽  
Lie Group ◽  
Closed Subgroup ◽  
Algebraic Groups ◽  
General Situation ◽  
Coset Space ◽  
Identity Component

In this paper we will be concerned with orbits of a closed subgroup Z of an algebraic group (respectively Lie group) G on a homogeneous space X for G. More precisely, let D be a closed subgroup of G and let X denote the coset space G/D. Let S be a subgroup of G and let Z denote (GS)0 the identity component of GS, the centralizer of S in G. We consider the orbits of Z on XS, the set of fixed points of S on X. We also treat the more general situation in which S is an algebraic group (respectively Lie group) which acts on G by automorphisms and acts on X compatibly with the action of G; again we consider the orbits of (GS)0 on XS.

Abstract Plancherel (Trace) Formulas over Homogeneous Spaces of Compact Groups

2017 ◽  
Vol 60 (1) ◽  
pp. 111-121 ◽  
Author(s):  
Arash Ghaani Farashahi
Keyword(s):  
Hilbert Space ◽  
Operator Theory ◽  
Closed Subgroup ◽  
Homogeneous Spaces ◽  
Compact Groups ◽  
Theory Approach ◽  
Coset Space ◽  
Trace Formulas ◽  
Left Coset ◽  
Abstract Notion

AbstractThis paper introduces a unified operator theory approach to the abstract Plancherel (trace) formulas over homogeneous spaces of compact groups. LetGbe a compact group and letHbe a closed subgroup ofG. LetG/Hbe the left coset space ofHinGand letμbe the normalized G-invariant measure onG-Hassociated with Weil’s formula. Then we present a generalized abstract notion of Plancherel (trace) formula for the Hilbert spaceL2(G/H,μ).

scholarly journals On a Type of Subgroups of a Compact Lie Group

1951 ◽  
Vol 2 ◽  
pp. 1-15 ◽  
Author(s):  
Yozô Matsushima
Keyword(s):  
Lie Group ◽  
Product Space ◽  
Closed Subgroup ◽  
The Other ◽  
Dimensional Sphere ◽  
Compact Lie Group ◽  
Coset Space ◽  
Fibre Bundles ◽  
Other Hand ◽  
Interesting Paper

Let G be a connected compact Lie group and H a connected closed subgroup. Then H is an orientable submanifold of G and we may consider H as a cycle in G. In his interesting paper on the topology of group manifolds H. Samelson has proved that, if H is not homologous to 0, then the homology ring of the coset space G/H is isomorphic to the homology ring of a product space of odd dimensional spheres and the homology ring of G is isomorphic to that of the product of the spaces H and G/H. On the other hand, in a recent investigation of fibre bundles’ T. Kudo has shown that, if the homology ring of the coset space G/H is isomorphic to that of an odd dimensional sphere, then H is not homologous to 0.

scholarly journals Invariant Measures on Coset Spaces

1961 ◽  
Vol 5 (2) ◽  
pp. 80-85 ◽  
Author(s):  
S. Świerczkowski
Keyword(s):  
Closed Subgroup ◽  
Invariant Measures ◽  
Locally Compact Group ◽  
Coset Space ◽  
Modular Functions ◽  
Natural Topology ◽  
Left Coset ◽  
Generic Element ◽  
Baire Measure ◽  
Coset Spaces

In this note we consider measures on a left coset space G/H, where G is a locally compact group and H is a closed subgroup. We assume the natural topology in G/H and we denote the generic element of this space by xH (x∈G). Every element t∈G defines a homeomorphism of G/H given by t(xH) = (tx)H. A. Weil showed that a Baire measure on G/H invariant under all these homeomorphisms can exist only ifΔ(ξ) = δ(ξ) for each ξ ∈ H,where Δ(x), δ(ξ) denote the modular functions in G, H [6, pp. 42–45]. We shall devote our investigations to inherited measures on G/H (cf. [3] and the definition below) invariant under homeomorphisms belonging to a normal and closed subgroup T ⊂ G.

scholarly journals Invariant measures on double coset spaces

1965 ◽  
Vol 5 (4) ◽  
pp. 495-505 ◽  
Author(s):  
Teng-Sun Liu
Keyword(s):  
Compact Group ◽  
Closed Subgroup ◽  
Invariant Measures ◽  
Double Coset ◽  
Locally Compact Group ◽  
Coset Space ◽  
Canonical Mapping ◽  
Locally Compact ◽  
Coset Spaces ◽  
Image Position

Let G be a locally compact group with left invariant Haar measure m. Le H be a closed subgroup of G and K a compact group of G. Let R be the equivalence relation in G defined by (a, b)∈R if and if a = kbh for some k in K and h in H. We call E =G/R the double coset space of G modulo K and H. Donote by a the canonical mapping of G onto E. It can be shown that E is a locally compact space and α is continous and open Let N be the normalizer of K in G, i. e. .

Weighted estimates for commutators of Hausdorff operators on the Heisenberg group

2020 ◽  
pp. 39-62
Author(s):  
Nguyen Minh Chuong ◽  
◽  
Dao Van Duong ◽  
Nguyen Duc Duyet ◽  
◽  
...  
Keyword(s):  
Heisenberg Group ◽  
Weighted Estimates ◽  
Hausdorff Operators
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