Abstract view
Admissibility for a Class of Quasiregular Representations


Published:20071001
Printed: Oct 2007
Abstract
Given a semidirect product $G = N \rtimes H$ where $N$ is%%
nilpotent, connected, simply connected and normal in $G$ and where
$H$ is a vector group for which $\ad(\h)$ is completely reducible and
$\mathbf R$split, let $\tau$ denote the quasiregular representation of
$G$ in $L^2(N)$. An element $\psi \in L^2(N)$ is said to be admissible
if the wavelet transform $f \mapsto \langle f, \tau(\cdot)\psi\rangle$
defines an isometry from $L^2(N)$ into $L^2(G)$. In this paper we give
an explicit construction of admissible vectors in the case where $G$
is not unimodular and the stabilizers in $H$ of its action on $\hat N$
are almost everywhere trivial. In this situation we prove
orthogonality relations and we construct an explicit decomposition of
$L^2(G)$ into $G$invariant, multiplicityfree subspaces each of which
is the image of a wavelet transform . We also show that, with the
assumption of (almosteverywhere) trivial stabilizers,
nonunimodularity is necessary for the existence of admissible
vectors.