The behavior of Fourier transforms for nilpotent Lie groups

Abstract

We study weak analogues of the Paley-Wiener Theorem for both the scalar-valued and the operator-valued Fourier transforms on a nilpotent Lie group G G . Such theorems should assert that the appropriate Fourier transform of a function or distribution of compact support on G G extends to be “holomorphic” on an appropriate complexification of (a part of) G ^ \hat G . We prove the weak scalar-valued Paley-Wiener Theorem for some nilpotent Lie groups but show that it is false in general. We also prove a weak operator-valued Paley-Wiener Theorem for arbitrary nilpotent Lie groups, which in turn establishes the truth of a conjecture of Moss. Finally, we prove a conjecture about Dixmier-Douady invariants of continuous-trace subquotients of C ∗ ( G ) C^{*}(G) when G G is two-step nilpotent.