Journal of Geometric Analysis, ( ISI ), No (19), Year (2009-3) , Pages (541-552)

Title : ( On the Square Integrability of Quasi Regular Representation on Semidirect Product Groups )

Authors: Rajab Ali Kamyabi Gol , Aliakbar Arefijamaal ,

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Let H be a locally compact group and K be a locally compact abelian group. Also letG = H ×τ K denote the semidirect product group ofH andK, respectively. Then the unitary representation (U,L2(K)) on G defined by U(h, x)f (y) = δ(h) 1 2 f (τh −1(yx −1)) is called the quasi regular representation. The properties of this representation in the case K = (Rn,+), have been studied by many authors under some specific assumptions. In this paper we aim to consider a general case and extend some of these properties when K is an arbitrary locally compact abelian group. In particular we wish to show that the two conditions (i) δH ≡ 1, and (ii) the stabilizers Hω are compact for a.e. ω ∈  K; both are necessary for square integrability of U. Furthermore, we shall consider some sufficient conditions for the square integrability of U. Also, for the square integrability of subrepresentations of U, we will introduce a concrete form of the Duflo-Moore operator.

Keywords

Semidirect product · Fourier transform · Locally compact abelian (LCA) group · Square integrable
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@article{paperid:1010889,
author = {Kamyabi Gol, Rajab Ali and Aliakbar Arefijamaal},
title = {On the Square Integrability of Quasi Regular Representation on Semidirect Product Groups},
journal = {Journal of Geometric Analysis},
year = {2009},
number = {19},
month = {March},
issn = {1050-6926},
pages = {541--552},
numpages = {11},
keywords = {Semidirect product · Fourier transform · Locally compact abelian (LCA) group · Square integrable representation},
}

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%0 Journal Article
%T On the Square Integrability of Quasi Regular Representation on Semidirect Product Groups
%A Kamyabi Gol, Rajab Ali
%A Aliakbar Arefijamaal
%J Journal of Geometric Analysis
%@ 1050-6926
%D 2009

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