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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The behavior of Fourier transforms for
nilpotent Lie groups


Authors: Ronald L. Lipsman and Jonathan Rosenberg
Journal: Trans. Amer. Math. Soc. 348 (1996), 1031-1050
MSC (1991): Primary 22E27; Secondary 43A30, 44A12, 22D25
MathSciNet review: 1370646
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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 $G$. Such theorems should assert that the appropriate Fourier transform of a function or distribution of compact support on $G$ extends to be ``holomorphic'' on an appropriate complexification of (a part of) $\hat G$. 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 $C^{*}(G)$ when $G$ is two-step nilpotent.


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Additional Information

Ronald L. Lipsman
Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland 20742
Email: rll@math.umd.edu

Jonathan Rosenberg
Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland 20742
Email: jmr@math.umd.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-96-01583-8
PII: S 0002-9947(96)01583-8
Keywords: Nilpotent Lie group, scalar-valued Fourier transform, operator-valued Fourier transform, Paley-Wiener Theorem, Radon transform, Dixmier-Douady \ class
Received by editor(s): September 4, 1994
Additional Notes: Both authors were partially supported by NSF grant DMS-92-25063.
Article copyright: © Copyright 1996 American Mathematical Society