Square-integrable Representations Research Articles

The Density Theorem for Operator-Valued Frames via Square-Integrable Representations of Locally Compact Groups

The Density Theorem for Operator-Valued Frames via Square-Integrable Representations of Locally Compact Groups

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.

Harmonic Analysis on the Affine Group of the Plane

For any natural number n, the group $$G_n$$ of all invertible affine transformations of n-dimensional Euclidean space has, up to equivalence, just one square-integrable representation and the left regular representation of $$G_n$$ is a multiple of this square-integrable representation. We provide a concrete realization $$\sigma $$ of this distinguished representation in the two-dimensional case. We explicitly decompose the Hilbert space $$L^2(G_2)$$ as a direct sum of left-invariant closed subspaces on each of which the left regular representation acts as a representation equivalent to $$\sigma $$ .

Smooth lattice orbits of nilpotent groups and strict comparison of projections

This paper provides sufficient density conditions for the existence of smooth vectors generating a frame or Riesz sequence in the lattice orbit of a square-integrable projective representation of a nilpotent Lie group. The conditions involve the product of lattice co-volume and formal dimension, and complement Balian–Low type theorems for the non-existence of smooth frames and Riesz sequences at the critical density. The proof hinges on a connection between smooth lattice orbits and generators for an explicitly constructed finitely generated Hilbert C⁎-module. An important ingredient in the approach is that twisted group C⁎-algebras associated to finitely generated nilpotent groups have finite decomposition rank, hence finite nuclear dimension, which allows us to deduce that any matrix algebra over such a simple C⁎-algebra has strict comparison of projections.

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.

Gamma factors of intertwining periods and distinction for inner forms of GL(n)

Let F be a p-adic field, E be a quadratic extension of F, D be an F-central division algebra of odd index and let θ be the Galois involution attached to E/F. Set H=GL(m,D), G=GL(m,D⊗FE), and let P=MU be a standard parabolic subgroup of G. Let w be a Weyl involution stabilizing M and Mθw be the subgroup of M fixed by the involution θw:m↦θ(wmw). We denote by X(M)w,− the complex torus of w-anti-invariant unramified characters of M. Following the global methods of [30], we associate to a finite length representation σ of M and to a linear form L∈HomMθw(σ,C) a family of H-invariant linear forms called intertwining periods on IndPG(χσ) for χ∈X(M)w,−, which is meromorphic in the variable χ. Then we give sufficient conditions for some of these intertwining periods, namely the open intertwining periods studied in [13], to have singularities. By a local/global method, we also compute in terms of Asai gamma factors the proportionality constants involved in their functional equations with respect to certain intertwining operators. As a consequence, we classify distinguished unitary and ladder representations of G, extending respectively the results of [42] and [26] for D=F, which both relied at some crucial step on the theory of Bernstein-Zelevinsky derivatives. We make use of one of the main results of [12] which in the case of the group G asserts that the Jacquet-Langlands correspondence preserves distinction. Such a result is for essentially square-integrable representations, but our method in fact allows us to use it only for cuspidal representations of G.

On commutative homogeneous vector bundles attached to nilmanifolds

The notion of Gelfand pair (G, K) can be generalized by considering homogeneous vector bundles over G/K instead of the homogeneous space G/K and matrix-valued functions instead of scalar-valued functions. This gives the definition of commutative homogeneous vector bundles. Being a Gelfand pair is a necessary condition for being a commutative homogeneous vector bundle. In the case when G/K is a nilmanifold having square-integrable representations, a big family of commutative homogeneous vector bundles was determined in [Transform. Groups 24 (2019), no. 3, 887–911]. In this paper we complete that classification.

Discrete frames for [formula omitted] arising from tiling systems on [formula omitted

A discrete frame for L2(Rd) is a countable sequence {ej}j∈J in L2(Rd) together with real constants 0<A≤B<∞ such thatA‖f‖22≤∑j∈J|〈f,ej〉|2≤B‖f‖22, for all f∈L2(Rd). We present a method of sampling continuous frames, which arise from square-integrable representations of affine-type groups, to create discrete frames for high-dimensional signals. Our method relies on partitioning the ambient space by using a suitable “tiling system”. We provide all relevant details for constructions in the case of Mn(R)⋊GLn(R), although the methods discussed here are general and could be adapted to some other settings. Finally, we prove significantly improved frame bounds over the previously known construction for the case of n=2.

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.

Balian–Low Type Theorems on Homogeneous Groups

We prove strict necessary density conditions for coherent frames and Riesz sequences on homogeneous groups. Let N be a connected, simply connected nilpotent Lie group with a dilation structure (a homogeneous group) and let (π,Hπ) be an irreducible, square-integrable representation modulo the center Z(N) of N on a Hilbert space Hπ of formal dimension dπ. If g ∈ Hπ is an integrable vector and the set {π(λ)g : λ ∈ Λ} for a discrete subset Λ ⊆ N/Z(N) forms a frame for Hπ, then its density satisfies the strict inequality D − (Λ) > dπ, where D − (Λ) is the lower Beurling density. An analogous density condition D+(Λ) < dπ holds for a Riesz sequence in Hπ contained in the orbit of (π,Hπ). The proof is based on a deformation theorem for coherent systems, a universality result for p-frames and p-Riesz sequences, some results from Banach space theory, and tools from the analysis on homogeneous groups.

Classification of standard modules with linear periods

Suppose that F is a non-Archimedean local field of characteristic not 2 and D is a central division algebra over F. Let n be a positive integer. We show a classification modulo essentially square-integrable representations of standard modules of GLn(D) which have non-zero linear periods. By this classification, the conjecture of Prasad and Takloo-Bighash is reduced to the case of essentially square-integrable representations.

On Square-Integrable Representations of A Lie Group of 4-Dimensional Standard Filiform Lie Algebra

In this paper, we study irreducible unitary representations of a real standard filiform Lie group with dimension equals 4 with respect to its basis. To find this representations we apply the orbit method introduced by Kirillov. The corresponding orbit of this representation is genereric orbits of dimension 2. Furthermore, we show that obtained representation of this group is square-integrable. Moreover, in such case , we shall consider its Duflo-Moore operator as multiple of scalar identity operator. In our case that scalar is equal to one.

Test vectors for finite periods and base change

Let E/F be a quadratic extension of finite fields. By a result of Gow, an irreducible representation π of G=GLn(E) has at most one non-zero H-invariant vector, up to multiplication by scalars, when H is GLn(F) or U(n,E/F). If π does have an H-invariant vector it is said to be H-distinguished. It is known, from the work of Gow, that H-distinction is characterized by base change from U(n,E/F), due to Kawanaka, when H is GLn(F) (resp. from GLn(F), due to Shintani, when H is U(n,E/F)). Assuming π is generic and H-distinguished, we give an explicit description of the H-invariant vector in terms of the Bessel function of π. Let ψ be a non-degenerate character of NG/NH and let Bπ,ψ be the (normalized) Bessel function of π on the ψ-Whittaker model. For the H-averageWπ,ψ=1|H|∑h∈Hπ(h)Bπ,ψ of the Bessel function, we prove thatWπ,ψ(In)=dim⁡ρdim⁡π⋅|GLn(E)||GLn(F)||U(n,E/F)|, where ρ is the irreducible representation of U(n,E/F) (resp. GLn(F)) that base changes to π when H is GLn(F) (resp. U(n,E/F)). As an application we classify the members of a generic L-packet of SLn(E) that admit invariant vectors for SLn(F). Finally we prove a p-adic analogue of our result for square-integrable representations in terms of formal degrees by employing the formal degree conjecture of Hiraga-Ichino-Ikeda [19].

Square-integrable representations and multipliers

Abstract We observe a connection between the existence of square-integrable representations of a locally compact group G and the existence of nonzero translation invariant operators from its Fourier–Stieltjes algebra B ⁢ ( G ) {B(G)} into L 2 ⁢ ( G ) {L^{2}(G)} or, equivalently, from L 2 ⁢ ( G ) {L^{2}(G)} into its enveloping von Neumann algebra C * ⁢ ( G ) * * {C^{*}(G)^{**}} .

Orthonormal bases in the orbit of square-integrable representations of nilpotent Lie groups

Let G be a connected, simply connected nilpotent group and π be an irreducible unitary representation of G that is square-integrable modulo its center Z(G) on L2(Rd). We prove that under reasonably weak conditions on G and π there exist a discrete subset Γ of G/Z(G) and some (relatively) compact set F⊆Rd such that{|F|−1/2π(γ)1F|γ∈Γ} forms an orthonormal basis of L2(Rd). This construction generalizes the well-known example of Gabor orthonormal bases in time-frequency analysis.The main theorem covers graded Lie groups with one-dimensional center. In the presence of a rational structure, the set Γ can be chosen to be a uniform subgroup of G/Z.

On distinguished square-integrable representations for Galois pairs and a conjecture of Prasad

We prove an integral formula computing multiplicities of square-integrable representations relative to Galois pairs over $p$-adic fields and we apply this formula to verify two consequences of a conjecture of Dipendra Prasad. One concerns the exact computation of the multiplicity of the Steinberg representation and the other the invariance of multiplicities by transfer among inner forms.

Generalized fixed-point algebras for twisted C⁎-dynamical systems

We show that Ralf Meyer's method of constructing generalized fixed-point algebras for C⁎-dynamical systems via their square-integrable representations on Hilbert C⁎-modules works for twisted C⁎-dynamical systems. To do this, we introduce the category of twisted Hilbert C⁎-modules and prove that Meyer's bra-ket operators are morphisms in this category. Some non-trivial results that we can obtain are a twisted-equivariant version of Kasparov's Stabilization Theorem and the existence of a classifying category for the category of Hilbert modules over any fixed reduced twisted crossed product. Furthermore, an interesting connection is made to the Gabor-analytic method of constructing Hilbert modules over non-commutative tori by Franz Luef.

On the irreducibility of global descents for even unitary groups and its applications

In this paper, we prove the irreducibility of global descents for even unitary groups. More generally, through Fourier-Jacobi coefficients of automorphic forms, we give a bijection between a certain set of irreducible cuspidal automorphic representations of U ( n , n ) ( A ) \mathrm {U}(n,n)(\mathbb {A}) and a certain set of irreducible square-integrable automorphic representations of U ( 2 n , 2 n ) ( A ) \mathrm {U}(2n, 2n)(\mathbb {A}) . We also give three applications of the irreducibility of global descents. As a global application, we prove a rigidity theorem for irreducible generic cuspidal automorphic representations of U ( n , n ) \mathrm {U}(n,n) . Moreover, as a local application, we prove the irreducibility of explicit local descents for a couple of supercuspidal representations and a local converse theorem for generic representations in the case of U ( n , n ) \mathrm {U}(n,n) .

Linear dependency of translations and square-integrable representations

Let G be a locally compact group. We examine the problem of determining when nonzero functions in L2(G) have linearly independent left translations. In particular, we establish some results for the case when G has an irreducible, square-integrable, unitary representation. We apply these results to the special cases of the affine group, the shearlet group, and the Weyl–Heisenberg group. We also investigate the case when G has an abelian, closed subgroup of finite index.

Covariant KSGNS construction and quantum instruments

We study completely positive (CP) [Formula: see text]-sesquilinear-form-valued maps on a unital [Formula: see text]-algebra [Formula: see text], where the sesquilinear forms operate on a module over a [Formula: see text]-algebra [Formula: see text]. We also study the cases when either one or both of the algebras are von Neumann algebras. Moreover, we assume that the CP maps are covariant with respect to actions of a symmetry group. This allows us to view these maps as generalizations of covariant quantum instruments. We determine minimal covariant dilations (KSGNS constructions) for covariant CP maps to find necessary and sufficient conditions for a CP map to be extreme in convex subsets of normalized covariant CP maps. As a special case, we study covariant quantum observables and instruments whose value space is a transitive space of a unimodular type-I group. Finally, we discuss the case of instruments that are covariant with respect to a square-integrable representation.