Remote Access Transactions of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9947-96-01583-8
MathSciNet review: 1370646
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [Ando] S. Ando, Paley-Wiener type theorem for Heisenberg groups, Proc. Japan Acad. 52 (1976), 331--333. MR 55:5796
  • [ArnG] D. Arnal and S. Gutt, Décomposition de $L^{2}(G)$ et transformation de Fourier adaptée pour un groupe $G$ nilpotent, C. R. Acad. Sci. Paris Sér. I Math. 306 (1988), 25--28. MR 88m:22015
  • [Br] I. Brown, Dual topology of a nilpotent Lie group, Ann. Sci. École Norm. Sup. 6 (1973), 407--411. MR 50:4813
  • [CorGr] L. Corwin and F. P. Greenleaf, Representations of nilpotent Lie groups and their Applications, Part 1: Basic Theory and Examples, Cambridge Studies in Advanced Math., vol. 18, Cambridge Univ. Press, Cambridge, New York, 1990. MR 92b:22007
  • [Dix1] J. Dixmier, Sur le dual d'un groupe de Lie nilpotent, Bull. Sci. Math. 90 (1966), 113--118. MR 35:298
  • [Dix2] J. Dixmier, Enveloping Algebras, North-Holland Math. Library, vol. 14, North-Holland, Amsterdam, New York, Oxford, 1977. MR 58:16803b
  • [Echt] S. Echterhoff, Crossed products with continuous trace, Habilitationsschrift, Universität-Gesamthochschule Paderborn, Paderborn, 1993.
  • [Fell] J. M. G. Fell, The structure of algebras of operator fields, Acta Math. 106 (1961), 233--280. MR 29:1547
  • [Hel] S. Helgason, The Radon Transform, Progress in Math., vol. 5, Birkhäuser, Boston, Basel, Stuttgart, 1980. MR 83f:43012
  • [Joy] K. T. Joy, A description of the topology on the dual space of a nilpotent Lie group, Pacific J. Math. 112 (1984), 135--139. MR 85e:22013
  • [Kum] K. Kumahara, An analogue of the Paley-Wiener Theorem for the Heisenberg group, Proc. Japan Acad. 47 (1971), 491--494. MR 46:9244
  • [MoorWo] C. C. Moore and J. A. Wolf, Square integrable representations of nilpotent groups, Trans. Amer. Math. Soc. 185 (1973), 445--462. MR 49:3033
  • [Moss] J. D. Moss, Jr., A Paley-Wiener Theorem for selected nilpotent Lie groups, J. Funct. Anal. 114 (1993), 395--411. MR 56:11992
  • [Mum] D. Mumford, Algebraic Geometry, I: Complex Projective Varieties, Grundlehren der math. Wiss., vol. 221, Springer-Verlag, Berlin, Heidelberg, New York, 1976. MR 94e:22017
  • [Park] R. Park, A Paley-Wiener Theorem for all two- and three-step nilpotent Lie groups, Preprint, 1994.
  • [Ped1] N. V. Pedersen, On the symplectic structure of coadjoint orbits of (solvable) Lie groups and applications, I, Math. Ann. 281 (1988), 633--669. MR 90c:22024
  • [Ped2] N. V. Pedersen, Geometric quantization and the universal enveloping algebra of a nilpotent Lie group, Trans. Amer. Math. Soc. 315 (1989), 511--563. MR 90c:22026
  • [Ped3] N. V. Pedersen, Geometric quantization and nilpotent Lie groups: a collection of examples, Univ. of Copenhagen, Copenhagen, 1988.
  • [Ped4] N. V. Pedersen, Orbits and primitive ideals of solvable Lie algebras, Math. Ann. 298 (1994), 275--326. MR 94m:17009
  • [Puk] L. Pukanszky, On Kirillov's character formula, J. reine ang. Math. 311 (1979), 408--440. MR 81b:22014
  • [RaRo] I. Raeburn and J. Rosenberg, Crossed products of continuous-trace $C^*$-algebras by smooth actions, Trans. Amer. Math. Soc. 305 (1988), 1--45. MR 89e:46077
  • [Ros] J. Rosenberg, $C^*$-algebras and Mackey's theory of group representations, $C^*$-algebras: 1943--1993, a Fifty Year Celebration (R. S. Doran, ed.), Contemp. Math., vol. 167, Amer. Math. Soc., Providence, RI, 1994, pp. (151--181). MR 94:17
  • [ScSi] D. Scott and A. Sitaram, Some remarks on the Pompieu problem for groups, Proc. Amer. Math. Soc. 104 (1988), 1261--1266. MR 89g:43009
  • [Sev1] L. A. Sevastyanov, The Paley-Wiener Theorem for the Heisenberg group, Trudy Univ. Druzby Narod. 80 (9) (1976), 122--126. MR 58:28277
  • [Sev2] L. A. Sevastyanov, An analogue of the Paley-Wiener Theorem for a nilpotent Lie group, An analogue of the Paley-Wiener Theorem for metabelian Lie groups, Topological Spaces and Their Mappings, Latv. Gos. Univ. Riga, Riga, 1981, pp. (130--136, 178, 184--185). MR 84a:22024
  • [Shaf] I. Shafarevich, Basic Algebraic Geometry, transl. by K. A. Hirsch, Grundlehren der math. Wiss., vol. 213, Springer-Verlag, Berlin, Heidelberg, New York, 1974 and 1977. MR 51:3163

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 22E27, 43A30, 44A12, 22D25

Retrieve articles in all journals with MSC (1991): 22E27, 43A30, 44A12, 22D25


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: https://doi.org/10.1090/S0002-9947-96-01583-8
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

American Mathematical Society