Decompositions of Generalized Wavelet Representations

Let $N$ be a simply connected, connected nilpotent Lie group which admits a uniform subgroup $\Gamma.$ Let $\alpha$ be an automorphism of $N$ defined by $\alpha\left( \exp X\right) =\exp AX.$ We assume that the linear action of $A$ is diagonalizable and we do not assume that $N$ is commutative. Let $W$ be a unitary wavelet representation of the semi-direct product group $\left\langle \cup_{j\in\mathbb{Z}}\alpha^{j}\left( \Gamma\right) \right\rangle \rtimes\left\langle \alpha\right\rangle $ defined by $W\left( \gamma,1\right) =f\left( \gamma^{-1}x\right) $ and $W\left( 1,\alpha\right) =\left\vert \det A\right\vert ^{1/2}f\left( \alpha x\right) .$ We obtain a decomposition of $W$ into a direct integral of unitary representations. Moreover, we provide an explicit unitary operator intertwining the representations, a precise description of the representations occurring, the measure used in the direct integral decomposition and the support of the measure. We also study the irreducibility of the fiber representations occurring in the direct integral decomposition in various settings. We prove that in the case where $A$ is an expansive automorphism then the decomposition of $W$ is in fact a direct integral of unitary irreducible representations each occurring with infinite multiplicities if and only if $N$ is non-commutative. This work naturally extends results obtained by H. Lim, J. Packer and K. Taylor who obtained a direct integral decomposition of $W$ in the case where $N$ is commutative and the matrix $A$ is expansive, i.e. all eigenvalues have absolute values larger than one.