A Paley-Wiener Theorem for Selected Nilpotent Lie Groups

This paper concerns itself with the problem of generalizing to nilpotent Lie groups a weak form of the classical Paley-Wiener theorem for Rn The generalization is accomplished for a large subclass of nilpotent Lie groups, as well as for an interesting example not in this subclass. The paper also shows that if N is any connected, simply connected nilpotent Lie group, then almost all representations π in the support of the Plancherel measure may be induced from a single family of Vergne polarizations, with each π being modelled in L2 of the same fixed subspace of the Lie algebra of N.