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Admissibility for a Class of Quasiregular Representations

  Published:2007-10-01
 Printed: Oct 2007
  • Bradley N. Currey
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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, multiplicity-free subspaces each of which is the image of a wavelet transform . We also show that, with the assumption of (almost-everywhere) trivial stabilizers, non-unimodularity is necessary for the existence of admissible vectors.
MSC Classifications: 22E27, 22E30 show english descriptions Representations of nilpotent and solvable Lie groups (special orbital integrals, non-type I representations, etc.)
Analysis on real and complex Lie groups [See also 33C80, 43-XX]
22E27 - Representations of nilpotent and solvable Lie groups (special orbital integrals, non-type I representations, etc.)
22E30 - Analysis on real and complex Lie groups [See also 33C80, 43-XX]
 

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