Date of Award

1991

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

First Advisor

Leonard F. Richardson

Abstract

This dissertation arose from efforts to prove the following conjecture, which generalizes to nilpotent Lie groups a weak form of the classical Paley-Wiener theorem for $\IR\sp{n}$: Let N be any connected, simply connected nilpotent Lie group with unitary dual N, and let $\varphi\in L\sbsp{c}{\infty}(N)$. Suppose that there exists a subset E $\subset$ N of positive Plancherel measure such that $\\varphi(\pi)$ = 0 for all $\pi\in$ E, where $\\varphi(\pi)$ is the operator-valued Fourier transform of $\varphi$. Then $\varphi$ = 0 almost everywhere on N. The writer has been able to prove a slightly weakened form of the conjecture for a large subclass of nilpotent Lie groups, and the conjecture itself for several interesting examples that lie outside this subclass. Chapter 3 contains these proofs, which make use of certain special polarizations (maximal subordinate subalgebras) of the Lie algebras there considered. Chapter 2 explains how to construct such polarizations in any nilpotent Lie algebra. Chapter 1 provides background for the work undertaken in Chapters 2 and 3.

Pages

81

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