THE BEHAVIOR OF FOURIER-TRANSFORMS FOR NILPOTENT LIE-GROUPS

We study weak analogues of the Paley-Wiener Theorem for both the scala r-valued and the operator-valued Fourier transforms on a nilpotent Lie group G. Such theorems should assert that the appropriate Fourier tra nsform of a function or distribution of compact support on G extends t o be ''holomorphic'' on an appropriate complexification of (a part of) (G) over cap. We prove the weak scalar-valued Paley-Wiener Theorem fo r some nilpotent Lie groups but show that it is false in general. We a lso prove a weak operator-valued Paley-Wiener Theorem for arbitrary ni lpotent Lie groups, which in turn establishes the truth of a conjectur e of Moss. Finally, we prove a conjecture about Dixmier-Douady invaria nts of continuous-trace subquotients of C(G) when G is two-step nilpo tent.