Square-integrable Representations Research Articles

The $$\alpha $$ α -modulation transform: admissibility, coorbit theory and frames of compactly supported functions

The $$\alpha $$ -modulation transform is a time-frequency transform generated by square-integrable representations of the affine Weyl–Heisenberg group modulo suitable subgroups. In this paper we prove new conditions that guarantee the admissibility of a given window function. We also show that the generalized coorbit theory can be applied to this setting, assuming specific regularity of the windows. This then yields canonical constructions of Banach frames and atomic decompositions in $$\alpha $$ -modulation spaces. In particular, we prove the existence of compactly supported (in time domain) vectors that are admissible and satisfy all conditions within the coorbit machinery, which considerably go beyond known results.

On the formal degrees of square-integrable representations of odd special orthogonal and metaplectic groups

The formal degree conjecture relates the formal degree of an irreducible square-integrable representation of a reductive group over a local field to the special value of the adjoint $\gamma$-factor of its $L$-parameter. In this paper, we prove the formal degree conjecture for odd special orthogonal and metaplectic groups in the generic case, which combined with Arthur's work on the local Langlands correspondence implies the conjecture in full generality.

A remark on Jacquet-Langlands correspondence and invariant $s$

Let $F$ be a non-Archimedean local field, and let $G$ be an inner form of $\mathrm{GL}_N(F)$ with $N \ge 1$. Let $\boldsymbol{\mathrm{JL}}$ be the Jacquet--Langlands correspondence between $\mathrm{GL}_N(F)$ and $G$. In this paper, we compute the invariant $s$ associated with the essentially square-integrable representation $\boldsymbol{\mathrm{JL}}^{-1}(\rho)$ for a cuspidal representation $\rho$ of $G$ by using the recent results of Bushnell and Henniart, and we restate the second part of a theorem given by Deligne, Kazhdan, and Vigneras in terms of the invariant $s$. Moreover, by using the parametric degree, we present a proof of the first part of the theorem.

COMPOSITION FACTORS OF A CLASS OF INDUCED REPRESENTATIONS OF CLASSICAL -ADIC GROUPS

We study induced representations of the form $\unicode[STIX]{x1D6FF}_{1}\times \unicode[STIX]{x1D6FF}_{2}\rtimes \unicode[STIX]{x1D70E}$, where $\unicode[STIX]{x1D6FF}_{1},\unicode[STIX]{x1D6FF}_{2}$ are irreducible essentially square-integrable representations of general linear group and $\unicode[STIX]{x1D70E}$ is a strongly positive discrete series of classical $p$-adic group, which naturally appear in the nonunitary dual. For $\unicode[STIX]{x1D6FF}_{1}=\unicode[STIX]{x1D6FF}([\unicode[STIX]{x1D708}^{a}\unicode[STIX]{x1D70C}_{1},\unicode[STIX]{x1D708}^{b}\unicode[STIX]{x1D70C}_{1}])$ and $\unicode[STIX]{x1D6FF}_{2}=\unicode[STIX]{x1D6FF}([\unicode[STIX]{x1D708}^{c}\unicode[STIX]{x1D70C}_{2},\unicode[STIX]{x1D708}^{d}\unicode[STIX]{x1D70C}_{2}])$ with $a\geqslant 1$ and $c\geqslant 1$, we determine composition factors of such induced representation.

Homogeneous spaces and square-integrable representations

Homogeneous spaces and square-integrable representations

Coorbit spaces and wavelet coefficient decay over general dilation groups

We study continuous wavelet transforms associated to matrix dilation groups giving rise to an irreducible square-integrable quasi-regular representation on L 2 ( R d ) \textrm {L}^2(\mathbb {R}^d) . It turns out that these representations are integrable as well, with respect to a wide variety of weights, thus allowing to consistently quantify wavelet coefficient decay via coorbit space norms. We then show that these spaces always admit an atomic decomposition in terms of bandlimited Schwartz wavelets. We exhibit spaces of Schwartz functions contained in all coorbit spaces, and dense in most of them. We also present an example showing that for a consistent definition of coorbit spaces, the irreducibility requirement cannot be easily dispensed with. We then address the question of how to predict wavelet coefficient decay from vanishing moment assumptions. To this end, we introduce a new condition on the open dual orbit associated to a dilation group: If the orbit is temperately embedded, it is possible to derive rather general weighted mixed L p \textrm {L}^{p} -estimates for the wavelet coefficients from vanishing moment conditions on the wavelet and the analyzed function. These estimates have various applications: They provide very explicit admissibility conditions for wavelets and integrable vectors, as well as sufficient criteria for membership in coorbit spaces. As a further consequence, one obtains a transparent way of identifying elements of coorbit spaces with certain (cosets of) tempered distributions. We then show that, for every dilation group in dimension two, the associated dual orbit is temperately embedded. In particular, the general results derived in this paper apply to the shearlet group and its associated family of coorbit spaces, where they complement and generalize the known results.

Jacquet modules of strongly positive representations of the metaplectic group $\widetilde {Sp(n)}$

Strongly positive discrete series represent a particularly important class of irreducible square-integrable representations of p-adic groups. Indeed, these representations are used as basic building blocks in known constructions of general discrete series. In this paper, we explicitly describe Jacquet modules of strongly positive discrete series. The obtained description of Jacquet modules, which relies on the classification of strongly positive discrete series given in [10], is valid in both classical and metaplectic case. We expect that our results, besides being interesting by themselves, should be relevant to some potential applications in the theory of automorphic forms, where both representations of metaplectic groups and structure of Jacquet modules play an important part.

A Bessel identity for the theta correspondence

We establish a spectral identity between global Bessel distributions with respect to generic cuspidal representations of an odd orthogonal group and the metaplectic cover of a symplectic group which are related by the theta correspondence. We also provide analogous local identities for square-integrable representations.

Schatten–von Neumann norm inequalities for two-wavelet localization operators associated with β-Stockwell transforms

In this article we define two-wavelet localization operators corresponding to an irreducible and square-integrable representation of a locally compact Hausdorff group on a Hilbert space. The group structure admitting an irreducible and square-integrable representation which is related to β-Stockwell transform, that we shall use in this article β ∈ R have been introduced in Boggiatto et al. [P. Boggiatto, C. Fernandez, and A. Galbis, A group representation related to the Stockwell transform, Indiana Univ. Math. J. 58(5) (2009), pp. 2277–2296]. The Schatten–von Neumann norm inequalities of these two-wavelet localization operators are established. The traces and the trace class norm inequalities of the trace class two-wavelet localization operators are given.

Generalized Formal Degree

For a real or p-adic connected reductive group G, Harish-Chandra introduced the Plancherel measure on the tempered dual Ĝtemp and founded the Plancherel formula [7, 18] relating functions on G to functions on Ĝtemp. While the Plancherel measure is equal to the formal degree for square-integrable representations, there has been no similar interpretation for tempered but nonsquare-integrable representations because formal degree is not defined for them yet. This paper intends to generalize formal degree to all unitary representations. The traditional theory of formal degree deals only with square-integrable representations of G on Hilbert spaces because it involves an integral of matrix coefficients that diverges for nonsquare-integrable representations. Let Z be the center of G, and π an irreducible unitary action of G on a Hilbert space Vπ . When π is square-integrable,

Strongly positive representations of metaplectic groups

In this paper, we obtain a classification of irreducible strongly positive square-integrable genuine representations of metaplectic groups over p-adic fields, using a purely algebraic approach. Our results parallel those of Mœglin and Tadić for classical groups, but their work relies on certain conjectures. On the other hand, our results are complete and there are no additional conditions or hypotheses. The important point to note here is that our results and techniques can be used in the case of classical p-adic groups in a completely analogous manner.

Modulation Spaces of Symbols for Representations of Nilpotent Lie Groups

We investigate continuity properties of operators obtained as values of the Weyl correspondence constructed by Pedersen (Invent. Math. 118:1–36, 1994) for arbitrary irreducible representations of nilpotent Lie groups. To this end we introduce modulation spaces for such representations and establish some of their basic properties. The situation of square-integrable representations is particularly important and in the special case of the Schrodinger representation of the Heisenberg group we recover the classical modulation spaces used in the time-frequency analysis.

The Continuous Shearlet Transform in Arbitrary Space Dimensions

This paper is concerned with the generalization of the continuous shearlet transform to higher dimensions. Similar to the two-dimensional case, our approach is based on translations, anisotropic dilations and specific shear matrices. We show that the associated integral transform again originates from a square-integrable representation of a specific group, the full n-variate shearlet group. Moreover, we verify that by applying the coorbit theory, canonical scales of smoothness spaces and associated Banach frames can be derived. We also indicate how our transform can be used to characterize singularities in signals.

The unitarizability of the Aubert dual of strongly positive square integrable representations

Under a natural assumption, which holds in the generic case, we prove in this paper that, for the classical p-adic groups, the Aubert dual of an irreducible, strongly positive square-integrable representation (in the sense of Mœgl., in-Tadic classification), is unitarizable. In this way, for this important class of representations, we verify the conjecture which states that the Aubert involution preserves unitarity.

Two-Wavelet Localization Operators on Homogeneous Spaces and Their Traces

We define two-wavelet localization operators in the setting of homogeneous spaces. We prove that they are in the trace class S 1 and give a trace formula for them. Then we show that two-wavelet operators on locally compact and Hausdorff groups endowed with unitary and square-integrable representations, general Daubechies operators and two-wavelet multipliers can be seen as two-wavelet localization operators on appropriate homogeneous spaces. Thus we give a unifying view concerning the three classes of linear operators. We also show that two-wavelet localization operators on $$\mathbb{R}$$ , considered as a homogeneous space, under the action of the affine group U are two-wavelet multipliers

The Homogeneous Approximation Property and the Comparison Theorem for Coherent Frames

OCHENIG Abstract. We show that the homogeneous approximation property and the comparison theorem hold for arbitrary coherent frames. This observation answers some questions about the density of frames that are not covered by the theory of Balan, Casazza, Heil, and Landau. The proofs are a variation of the method developed by Ramanathan and Steger. Frames provide redundant non-orthogonal expansions in Hilbert space, intu- itively they should therefore be denser than orthonormal bases. The first funda- mental result is Landau's necessary condition for sets of sampling of band-limited functions (11). A second fundamental idea is the homogeneous approximation property and the comparison theorem of Ramanathan and Steger (12) in the con- text of Gabor frames. Since then many contributions have investigated the den- sity of frames and varied and applied the method of Ramanathan and Steger. See (3,4,8,13) for a sample of papers. The approach in (12) culminates in the deep investigation of Balan, Casazza, Heil, and Landau (1,2) who have found a density theory for a general class of frames that are labeled by a discrete Abelian group. However, the BCHL theory is not universally applicable, for instance, the density of wavelet frames does not fall under this general theory, and a separate investiga- tion is necessary, as was shown in a sequence of papers by Heil and Kutyniok (9,10) and by Sun (14-16). Their work uses quite explicit and long calculations in the group of affine transformations. In this note we show that the approach of Ramanathan and Steger yields the homogeneous approximation property and a comparison theorem for the class of coherent frames. These frames arise as subsets of the orbit of a square-integrable group representation. In particular, wavelet frames possess this structure. The advantage of the abstract point of view is the conciseness and simplicity of the proofs and the much more general range of validity. The additional insight is that the homogeneous approximation property and the comparison theorem are a consequence of structure alone, we do not need to assume additional localization properties as in (1). We will work in the context of a locally compact group G with Haar measure dx. We write |U | = R G χU(x) dx for the measure of a set U ⊆ G and K c for the complement of K in G.

Unitarizability of a certain class of irreducible representations of classical groups

We prove that a certain class of irreducible representations of the classical p-adic groups is unitarizable and in general, can be isolated in the unitary dual. These representations are Aubert duals of a certain class of square-integrable representations, thus, in this case, Bernstein’s conjecture, which states that the Aubert involution preserves unitarizability, is confirmed.

Polar Wavelet Transforms and Localization Operators

The notion of a polar wavelet transform is introduced. The underlying non-unimodular Lie group, the associated square-integrable representations and admissible wavelets are studied. The resolution of the identity formula for the polar wavelet transform is then formulated and proved. Localization operators corresponding to the polar wavelet transforms are then defined. It is proved that under suitable conditions on the symbols, the localization operators are, in descending order of complexity, paracommutators, paraproducts and Fourier multipliers.

Distinguished representations, base change, and reducibility for unitary groups

We show the equality of the local Asai L-functions defined via the Rankin-Selberg method and the Langlands-Shahidi method for a square-integrable representation of GL n (E). As a consequence, we characterize reducibility of certain induced representations of U(n,n), and the image of the base change map from U(n) to GL n (E) in terms of GL n (F)-distinguishedness. We also determine precisely when the Steinberg representation of GL n (E) is distinguished with respect to a character of F ∗ .

Wavelet Frames on Groups and Hypergroups via Discretization of Calder�n Formulas

Continuous wavelets are often studied in the general framework of representation theory of square-integrable representations, or by using convolution relations and Fourier transforms. We consider the well-known problem whether these continuous wavelets can be discretized to yield wavelet frames. In this paper we use Calderon-Zygmund singular integral operators and atomic decompositions on spaces of homogeneous type, endowed with families of general translations and dilations, to attack this problem, and obtain strong convergence results for wavelets expansions in a variety of classical functional spaces and smooth molecule spaces. This approach is powerful enough to yield, in a uniform way, for example, frames of smooth wavelets for matrix dilations in ℝ n , for an affine extension of the Heisenberg group, and on many commutative hypergroups.