Quasi-regular Representation Research Articles

Integrability properties of quasi-regular representations ofNAgroups

Let $G = N \rtimes A$, where $N$ is a graded group and $A = \mathbb{R}^+$ acts on $N$ via homogeneous dilations. The quasi-regular representation $\pi = \mathrm{ind}_A^G (1)$ of $G$ can be realised to act on $L^2 (N)$. It is shown that for a class of analysing vectors the associated wavelet transform defines an isometry from $L^2 (N)$ into $L^2 (G)$ and that the integral kernel of the corresponding orthogonal projector has polynomial off-diagonal decay. The obtained reproducing formula is instrumental for proving decomposition theorems for function spaces on nilpotent Lie groups.

Boundary maps, germs and quasi-regular representations

We investigate the tracial and ideal structures of C⁎-algebras of quasi-regular representations of stabilizers of boundary actions.Our main tool is the notion of boundary maps, namely Γ-equivariant unital completely positive maps from Γ-C⁎-algebras to C(∂FΓ), where ∂FΓ denotes the Furstenberg boundary of a group Γ.For a unitary representation π coming from the groupoid of germs of a boundary action, we show that there is a unique boundary map on Cπ⁎(Γ). Consequently, we completely describe the tracial structure of the C⁎-algebras Cπ⁎(Γ), and for any Γ-boundary X, we completely characterize the simplicity of the C⁎-algebras generated by the quasi-regular representations λΓ/Γx associated to stabilizer subgroups Γx for any x∈X.As an application, we show that the C⁎-algebra generated by the quasi-regular representation λT/F associated to Thompson's groups F≤T does not admit traces and is simple.

Coorbit spaces associated to integrably admissible dilation groups

This paper considers coorbit spaces parametrized by mixed, weighted Lebesgue spaces with respect to the quasi-regular representation of the semi-direct product of Euclidean space and a suitable matrix dilation group. The class of dilation groups that we allow, the so-called integrably admissible dilation groups, contains the matrix groups yielding an irreducible, square-integrable quasi-regular representation as a proper subclass. The obtained scale of coorbit spaces extends therefore the well-studied wavelet coorbit spaces associated to discrete series representations. We show that for any integrably admissible dilation group there exists a convienent space of smooth, admissible analyzing vectors that can be used to define a consistent coorbit space possessing all the essential properties that are known to hold in the setting of discrete series representations. In particular, the obtained coorbit spaces can be realized as Besov-type decomposition spaces by means of a Littlewood—Paley-type characterization. The classes of anisotropic Besov spaces associated to expansive matrices are shown to coincide precisely with the coorbit spaces induced by the integrably admissible one-parameter groups.

Unitarization of the Horocyclic Radon Transform on Homogeneous Trees

Following previous work in the continuous setup, we construct the unitarization of the horocyclic Radon transform on a homogeneous tree X and we show that it intertwines the quasi regular representations of the group of isometries of X on the tree itself and on the space of horocycles.

On the oriented Thompson subgroup F→3 and its relatives in higher Brown–Thompson groups

A few years ago the so-called oriented subgroup [Formula: see text] of the Thompson group [Formula: see text] was introduced by V. Jones while investigating the connections between subfactors and conformal field theories. In the coding of links and knots by elements of [Formula: see text] it corresponds exactly to the oriented ones. Thanks to the work of Golan and Sapir, [Formula: see text] provided the first example of a maximal subgroup of infinite index in [Formula: see text] different from the parabolic subgroups that fix a point in [Formula: see text]. In this paper we investigate possible analogues of [Formula: see text] in higher Thompson groups [Formula: see text], with [Formula: see text], introduced by Brown. Most notably, we study algebraic properties of the oriented subgroup [Formula: see text] of [Formula: see text], as described recently by Jones, and prove in particular that it gives rise to a non-parabolic maximal subgroup of infinite index in [Formula: see text] and that the corresponding quasi-regular representation is irreducible.

On double cosets of groups GL (n) with respect to subgroups of block strictly triangular matrices

ABSTRACT We parametrize the space of double cosets of the group with respect to two subgroups , of block strictly triangular matrices. In the appendix, we consider the quasi-regular representation of in on , observe that it admits an additional group of symmetries, find the joint spectrum, and observe that it is multiplicity free.

Generalized quasi-regular representation and its applications for shearlet transforms

The construction of continuous shearlet transform has been extended to higher dimensions. It was generalized to a group that is topologically isomorphic to a group of semidirect product of locally compact groups. In this paper, by a unified theoretical linear algebra approach to the representation theory, a class of continuous shearlet transforms obtained from the generalized quasi-regular representation is presented. In order to develop such representation, we utilize a homogeneous space with a relatively invariant Radon measure as tool from computational and abstract harmonic analysis.

Duflo-Moore Operator for The Square-Integrable Representation of 2-Dimensional Affine Lie Group

In this paper, we study the quasi-regular and the irreducible unitary representation of affine Lie group of dimension two. First, we prove a sharpening of Fuhr’s work of Fourier transform of quasi-regular representation of . The second, in such the representation of affine Lie group is square-integrable then we compute its Duflo-Moore operator instead of using Fourier transform as in F hr’s work.

Existence of representation frames based on wave packet groups

Let $H$ be a locally compact group, $K$ a locally compact abelian group with dual group $\hat{K}$. In this article, we consider the wave packet group $G_{\Theta}$ which is the semidirect product of locally compact groups $H$ and $K\times \hat{K}$, where $\Theta$ is a continuous homomorphism from $H$ into $Aut(K\times\hat{K})$. We review the quasi regular representation on $G_{\Theta}$ and extend the continuous Zak transform to $L^{2}(G_{\Theta})$. Moreover, we state a continuous frame based on $G_{\Theta}$ to reconstruct the element of $L^{2}\left(K\times \hat{K}\right)$. These results are extended to more general wave packet groups. Finally, we establish some methods to find dual of such continuous frames in the form of original frames.

UNIFORM KAZHDAN CONSTANTS AND PARADOXES OF THE AFFINE PLANE

Let G = SL(2, ℤ) ⋉ ℤ2 and H = SL(2, ℤ). We prove that the action G ↷ ℝ2 is uniformly non-amenable and that the quasi-regular representation of G on ℓ2(G/H) has a uniform spectral gap. Both results are a consequence of a uniform quantitative form of ping-pong for affine transformations, which we establish here.

Topological Boundaries of Unitary Representations

AbstractWe introduce and study a generalization of the notion of the Furstenberg boundary of a discrete group $\Gamma $ to the setting of a general unitary representation $\pi : \Gamma \to B(\mathcal H_\pi )$. This space, which we call the “Furstenberg–Hamana boundary” (or "FH-boundary") of the pair $(\Gamma , \pi )$, is a $\Gamma $-invariant subspace of $B(\mathcal H_\pi )$ that carries a canonical $C^{\ast }$-algebra structure. In many natural cases, including when $\pi $ is a quasi-regular representation, the Furstenberg–Hamana boundary of $\pi $ is commutative but can be noncommutative in general. We study various properties of this boundary and discuss possible applications, for example in uniqueness of certain types of traces.

Quasi-regular representations of discrete groups and associated $C^*$-algebras

Let G be a countable group. We introduce several equivalence relations on the set Sub(G) of subgroups of G, defined by properties of the quasi-regular representations λ G/H associated to H ∈ Sub(G) and compare them to the relation of G-conjugacy of subgroups. We define a class Sub sg (G) of subgroups (these are subgroups with a certain spectral gap property) and show that they are rigid, in the sense that the equivalence class of H ∈ Sub sg (G) for any one of the above equivalence relations coincides with the G-conjugacy class of H. Next, we introduce a second class Sub w−par (G) of subgroups (these are subgroups which are weakly parabolic in some sense) and we establish results concerning the ideal structure of the C

On a Family of Representations of Residually Finite Groups

For a residually finite group G, its normal subgroups G ⊃ G1 ⊃ G2⋯ with $\cap _{n\in \mathbb N}G_{n}=\{e\}$ and for a growth function γ we construct a unitary representation πγ of G. For the minimal growth, πγ is weakly equivalent to the regular representation, and for the maximal growth it is weakly equivalent to the direct sum of the quasiregular representations on the quotients G/Gn. In the case of intermediate growth we show two examples of different behaviour of πγ.

The Radon transform intertwines wavelets and shearlets

We prove that the unitary affine Radon transform intertwines the quasi-regular representation of a class of semidirect products, built by shearlet dilation groups and translations, and the tensor product of a standard wavelet representation with a wavelet-like representation. This yields a formula for shearlet coefficients that involves only integral transforms applied to the affine Radon transform of the signal, thereby opening new perspectives in the inversion of the Radon transform.

Ergodic boundary representations

We prove a von Neumann-type ergodic theorem for averages of unitary operators arising from the Furstenberg–Poisson boundary representation (the quasi-regular representation) of any lattice in a non-compact connected semisimple Lie group with finite center.

C*-simplicity and the unique trace property for discrete groups

A discrete group is said to be C*-simple if its reduced C*-algebra is simple, and is said to have the unique trace property if its reduced C*-algebra has a unique tracial state. A dynamical characterization of C*-simplicity was recently obtained by the second and third named authors. In this paper, we introduce new methods for working with group and crossed product C*-algebras that allow us to take the study of C*-simplicity a step further, and in addition to settle the longstanding open problem of characterizing groups with the unique trace property. We give a new and self-contained proof of the aforementioned characterization of C*-simplicity. This yields a new characterization of C*-simplicity in terms of the weak containment of quasi-regular representations. We introduce a convenient algebraic condition that implies C*-simplicity, and show that this condition is satisfied by a vast class of groups, encompassing virtually all previously known examples as well as many new ones. We also settle a question of Skandalis and de la Harpe on the simplicity of reduced crossed products. Finally, we introduce a new property for discrete groups that is closely related to C*-simplicity, and use it to prove a broad generalization of a theorem of Zimmer, originally conjectured by Connes and Sullivan, about amenable actions.

A classification of continuous wavelet transforms in dimension three

This paper presents a full catalogue, up to conjugacy and subgroups of finite index, of all matrix groups H < GL ( 3 , R ) that give rise to a continuous wavelet transform with associated irreducible quasi-regular representation. For each group in this class, coorbit theory allows to consistently define spaces of sparse signals, and to construct atomic decompositions converging simultaneously in a whole range of these spaces. As an application of the classification, we investigate the existence of compactly supported admissible vectors and atoms for the groups.

Study of shearlet transform using block matrix dilation

In this paper, the shearlet theory is extended from ordinary matrices to block matrices and its properties are investigated. More precisely, the block shearlet group is defined as a semidirect product group and the block shearlet transform is constructed by means of a quasiregular representation of this semidirect product group. In addition, we show that the block shearlet transform can overcome two major issues of the classical shearlet transform: time complexity and redundancy. We also report the result of careful experiments on video denoising which compares our proposed method with state of the art multiscale techniques, including curvelets and 3D shearlet transform.

An ergodic theorem for the quasi-regular representation of the free group

We prove the weak-$*$ convergence of a certain sequence of averages of unitary operators associated to the action of the free group on its Gromov boundary. This result, which can be thought as an ergodic theorem a la von Neumann with coefficients, provides a new proof of the irreducibility of the quasi-regular representation of the free group.

On spectra of Koopman, groupoid and quasi-regular representations

In this paper we investigate relations between Koopman, groupoid and quasi-regular representations of countable groups. We show that for an ergodic measure class preserving action of a countable group G on a standard Borel space the associated groupoid and quasi-regular representations are weakly equivalent and weakly contained in the Koopman representation. Moreover, if the action is hyperfinite then the Koopman representation is weakly equivalent to the groupoid. As a corollary of our results we obtain a continuum of pairwise disjoint pairwise equivalent irreducible representations of weakly branch groups. As an illustration we calculate spectra of regular, Koopman and groupoid representations associated to the action of the 2-group of intermediate growth constructed by the second author in 1980.