Square-Integrability of Induced Representations of Semidirect Products

Abstract

We consider a semidirect product G=A×′H, with A abelian, and its unitary representations of the form [Formula: see text] where x0 is in the dual group of A, G0 is the stability group of x0 and m is an irreducible unitary representation of G0∩H. We give a new selfcontained proof of the following result: the induced representation [Formula: see text] is square-integrable if and only if the orbit G[x0] has nonzero Haar measure and m is square-integrable. Moreover we give an explicit form for the formal degree of [Formula: see text].

Analytic Features of Reproducing Groups for the Metaplectic Representation

“…In Section 5 we establish some connections with wavelet theory. We exhibit another reproducing subgroup of Sp(2, R), which is a covering of the similitude group of the plane SI M (2). We then show that our theory, for both T DS (2) and SI M (2), parallels the theory developed in the context of two-dimensional wavelets.…”

“…They admit a natural unitary representation on L 2 (R d ), the main ingredient for the construction of a wavelet transform. Initially, only irreducible square-integrable representations were considered [2,12], but it soon became clear that nonirreducible representations [15,19,13] are of relevance as well.…”

“…Some of these appear in [7]; a deeper study of the geometry of admissible groups is developed in a forthcoming article [6]. In Section 4, Theorem 1 is applied to prove the admissibility of a subgroup H of Sp(2, R) that we denote T DS (2). In Section 5 we establish some connections with wavelet theory.…”

Wavelet Transforms for Semidirect Product Groups with Not Necessarily Commutative Normal Subgroups

“…The results can be naturally generalized to the case that G is the semidirect product group of a vector group V with a linear group H on V (see [5,8] and [10]), and further extensions of the theory have been developed in various directions. For example, wavelet transforms associated to nonirreducible representations are considered recently by [12] and [19], while wavelets for vector-valued functions associated to induced representations are studied by [2] and [3] (see also [1,Chapter 10]). Discretizations of the theory are considered by many authors (see [4,15,16] for example).…”

“…Furthermore, if µ(N \ k∈K O * λ k ) = 0, the unitary representation (L, L 2 (N )) is decomposed into the direct sum of such subrepresentations (Proposition 5). When φ is an element of the irreducible subspace, one has φ 2 = O * λ k π λ (φ) 2 HS dµ(λ) for some k. Then φ is admissible if and only if the integral…”