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

Interview with John Dickson

An interview with John Dickson regarding his experiences with Edward Lansdale in the Philippines.https://scholars.fhsu.edu/lansdale_interviews/1007/thumbnail.jp

Interview with Dorothy Bohannan

An interview with Dorothy Bohannan regarding her experiences as the wife of a CIA operative in the Philippines leading up to the Vietnam War.https://scholars.fhsu.edu/lansdale_interviews/1004/thumbnail.jp

November 28, 1984 Lecture by Edward Lansdale

A lecture given by Edward Lansdale regarding his life experiences.https://scholars.fhsu.edu/lansdale_interviews/1041/thumbnail.jp

Interview with Spence Davis

A transcript of an interview with Spence Davis regarding his experiences with Edward Lansdale in the Philippines. The audio for this interview has been lost.https://scholars.fhsu.edu/lansdale_interviews/1009/thumbnail.jp

Interview with Manny Chavez

An interview with Manny Chavez regarding his experiences during the Vietnam War Era.https://scholars.fhsu.edu/lansdale_interviews/1008/thumbnail.jp

Interview with Peter Lansdale

An interview with Peter Lansdale regarding his experiences with Edward Lansdale.https://scholars.fhsu.edu/lansdale_interviews/1015/thumbnail.jp

Interview with Joseph Baker

An interview with Joseph Baker regarding his experiences with Edward Lansdale and the intelligence community.https://scholars.fhsu.edu/lansdale_interviews/1001/thumbnail.jp

December 19, 1984 Interview with Edward Lansdale

An interview with Edward Lansdale regarding his experiences in Vietnam.https://scholars.fhsu.edu/lansdale_interviews/1055/thumbnail.jp

Interview with former director of the CIA William Colby

An interview with former director of the CIA William Colby regarding his experiences leading up to and during the Vietnam War Era. Colby was the Director of Central Intelligence from 1973-1976.https://scholars.fhsu.edu/lansdale_interviews/1005/thumbnail.jp