A note on the L 2 −harmonic analysis of the Joint-Eigenspace Fourier transform (original) (raw)

On the Fourier transform of a compact semisimple Lie group

Journal of the Australian Mathematical Society, 1994

We develop a concrete Fourier transform on a compact Lie group by means of a symbol calculus, or *-product, on each integral co-adjoint orbit. These *-products are constructed by means of a moment map defined for each irreducible representation. We derive integral formulae for these algebra structures and discuss the relationship between two naturally occurring inner products on them. A global Kirillov-type character is obtained for each irreducible representation. The case of SU(2) is treated in some detail, where some interesting connections with classical spherical trigonometry are obtained.

Fourier-Stieltjes transform defined by induced representation on locally compact groups

2022

Abstract. In this work we extend the Fourier-Stieltjes transform of a vector measure and a continuous function defined on compact groups to locally compact groups. To do so, we consider a representation L of a normal compact subgroup K of a locally compact group G, and we use a representation of G induced by that of L. Then, we define the Fourier-Stieltjes transform of a vector measure and that of a continuous function with compact support defined on G from the representation of G. Then, we extend the Shur orthogonality relation established for compact groups to locally compact groups by using the representations of G induced by the unitary representations of one of its normal compact subgroups. This extension enables us to develop a Fourier-Stieltjes transform in series form that is linear, continuous, and invertible.

The behavior of Fourier transforms for nilpotent Lie groups

Transactions of the American Mathematical Society, 1996

We study weak analogues of the Paley-Wiener Theorem for both the scalar-valued and the operator-valued Fourier transforms on a nilpotent Lie group G G . Such theorems should assert that the appropriate Fourier transform of a function or distribution of compact support on G G extends to be “holomorphic” on an appropriate complexification of (a part of) G ^ \hat G . We prove the weak scalar-valued Paley-Wiener Theorem for some nilpotent Lie groups but show that it is false in general. We also prove a weak operator-valued Paley-Wiener Theorem for arbitrary nilpotent Lie groups, which in turn establishes the truth of a conjecture of Moss. Finally, we prove a conjecture about Dixmier-Douady invariants of continuous-trace subquotients of C ∗ ( G ) C^{*}(G) when G G is two-step nilpotent.

On harmonic analysis of spherical convolutions on semisimple Lie groups

This paper contains a non-trivial generalization of the Harish-Chandra transforms on a connected semisimple Lie group G, with finite center, into what we term spherical convolutions. Among other results we show that its integral over the collection of bounded spherical functions at the identity element e ∈ G is a weighted Fourier transforms of the Abel transform at 0. Being a function on G, the restriction of this integral of its spherical Fourier transforms to the positive-definite spherical functions is then shown to be (the non-zero constant multiple of) a positive-definite distribution on G, which is tempered and invariant on G = SL(2, R). These results suggest the consideration of a calculus on the Schwartz algebras of spherical functions. The Plancherel measure of the spherical convolutions is also explicitly computed.

An Lp−Lq version of Hardy's theorem for spherical Fourier transform on semisimple Lie groups

International Journal of Mathematics and Mathematical Sciences, 2004

We consider a real semisimple Lie groupGwith finite center andKa maximal compact subgroup ofG. We prove anLp−Lqversion of Hardy's theorem for the spherical Fourier transform onG. More precisely, leta,bbe positive real numbers,1≤p,q≤∞, andfaK-bi-invariant measurable function onGsuch thatha−1f∈Lp(G)andeb‖λ‖2ℱ(f)∈Lq(𝔞+*)(hais the heat kernel onG). We establish that ifab≥1/4andporqis finite, thenf=0almost everywhere. Ifab<1/4, we prove that for allp,q, there are infinitely many nonzero functionsfand ifab=1/4withp=q=∞, we havef=const ha.

Characters, bi-modules and representations in Lie group harmonic analysis

This paper is a personal look at some issues in the representation theory of Lie groups having to do with the role of commutative hypergroups, bi-modules, and the construction of representations. We begin by considering Frobenius' original approach to the character theory of a finite group and extending it to the Lie group setting, and then introduce bi-modules as objects intermediate between characters and representations in the theory. A simplified way of understanding the formalism of geometric quantization, at least for compact Lie groups, is presented, which leads to a canonical bi-module of functions on an integral coadjoint orbit. Some meta-mathematical issues relating to the construction of representations are considered.

Regular representations of time-frequency groups

Mathematische Nachrichten, 2014

In this paper, we study the Plancherel measure of a class of non-connected nilpotent groups which is of special interest in Gabor theory. Let G be a time-frequency group. More precisely, that is G = T k , M l : k ∈ Z d , l ∈ BZ d , T k , M l are translations and modulations operators acting in L 2 (R d ), and B is a non-singular matrix. We compute the Plancherel measure of the left regular representation of G which is denoted by L. The action of G on L 2 (R d ) induces a representation which we call a Gabor representation. Motivated by the admissibility of this representation, we compute the decomposition of L into direct integral of irreducible representations by providing a precise description of the unitary dual and its Plancherel measure. As a result, we generalize Hartmut Führ's results which are only obtained for the restricted case where d = 1, B = 1/L, L ∈ Z and L > 1. Even in the case where G is not type I, we are able to obtain a decomposition of the left regular representation of G into a direct integral decomposition of irreducible representations when d = 1. Some interesting applications to Gabor theory are given as well. For example, when B is an integral matrix, we are able to obtain a direct integral decomposition of the Gabor representation of G.

The Fourier transform in quantum group theory

2006

The Fourier transform, known in classical analysis, and generalized in abstract harmonic analysis, can also be considered in the theory of locally compact quantum groups. In this note, I discuss some aspects of this more general Fourier transform. In order to avoid technical difficulties, typical for the analytical approach, I will restrict to the algebraic quantum groups. Roughly speaking, these are the locally compact quantum groups that can be treated with purely algebraic methods (in the framework of multiplier Hopf algebras). I will illustrate various notions and results using not only classical Fourier theory on the circle group BbbT\Bbb TBbbT, but also on the additive group BbbQp\Bbb Q_pBbbQp of ppp-adic numbers. It should be observed however that these cases are still too simple to illustrate the full power of the more general theory.