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Homogeneous distributions on the Heisenberg group and representations of $ {\rm SU}(2,1)$


Author: R. C. Fabec
Journal: Trans. Amer. Math. Soc. 328 (1991), 351-391
MSC: Primary 22E25; Secondary 22E45, 22E46
DOI: https://doi.org/10.1090/S0002-9947-1991-1043858-7
MathSciNet review: 1043858
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Abstract: A 'Fourier' transform of tempered distributions on the Heisenberg group is defined to analyze homogeneous distributions relative the group of dilations $ (z,t) \mapsto (r\,z,{r^2}t)$, $ r \in {\mathbf{R}}$. An inversion formula is derived for the abelian central Fourier transform of the distribution. These formulas are applied to the family of homogeneous distributions defining the intertwining operators for the group $ {\text{SU}}(2,1)$. Explicit unitary structures are determined on subquotient representations and their spectral decompositions on the minimal parabolic subgroup are obtained.


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

  • [1] A. Abramowitz and I. A. Stegun, Handbook of mathematical functions, National Bureau of Standards, Washington, D.C., 1964.
  • [2] L. Corwin and F. P. Greenleaf, Fourier transforms of smooth functions on certain nilpotent Lie groups, J. Funct. Anal. 37 (1980), 203-217. MR 578932 (81f:22012)
  • [3] M. Cowling and A. Koranyi, Harmonic analysis on Heisenberg type groups from a geometric viewpoint, Lie Groups Representations. III, Lecture Notes in Math., vol. 1077, Springer, 1984, pp. 60-100. MR 765552 (86g:22013)
  • [4] M. Cowling, Harmonic analysis on some nilpotent groups, Topics in Modern Analysis, Vols. I, II, Turin/Milan, 1982, pp. 81-123. MR 748862 (85i:22012)
  • [5] R. Fabec, Localizable representations of the De Sitter group, J. Analyse Math. 35 (1979), 151-207. MR 555303 (81e:22020)
  • [6] R. Howe, Quantum mechanics and partial differential equations, J. Funct. Anal. 38 (1980), 188-254. MR 587908 (83b:35166)
  • [7] A. W. Knapp and E. M. Stein, Intertwining operators for semisimple groups, Ann. of Math. (2) 93 (1971), 489-578. MR 0460543 (57:536)
  • [8] H. Kraljevic, Representations of the universal covering group $ {\text{SU}}\,(n,1)$, Glas. Mat. Ser. III 8 (28) (1973), 23-72. MR 0330355 (48:8692)
  • [9] -, On representations of the group $ {\text{SU}}\,(n,1)$, Trans. Amer. Math. Soc. 221 (1976), 433-448. MR 0409725 (53:13477)
  • [10] R. A. Kunze and E. M. Stein, Uniformly bounded representations and harmonic analysis on the $ 2 \times 2$ real unimodular group, Amer. J. Math. 82 (1960), 1-62. MR 0163988 (29:1287)
  • [11] -, Uniformly bounded representations, II. Analytic continuation of the principal series of representations of the $ n \times n$ complex unimodular group, Amer. J. Math. 83 (1961), 723-786. MR 0163989 (29:1288)
  • [12] -, Uniformly bounded representations, III. Intertwining operators for the principal series on semisimple groups, Amer. J. Math. 83 (1967), 385-442. MR 0231943 (38:269)
  • [13] R. J. Lipsman, Uniformly bounded representations of the Lorentz groups, Amer. J. Math. 91 (1969), 983-962. MR 0267044 (42:1946)
  • [14] G. W. Mackey, Unitary representations of group extensions. I, Acta Math. 99 (1958), 265-311. MR 0098328 (20:4789)
  • [15] C. Moore and J. Wolf, Square integrable representations of nilpotent Lie groups, Trans. Amer. Math. Soc. 185 (1973), 445-462. MR 0338267 (49:3033)
  • [16] P. J. Sally, Jr., Analytic continuation of the irreducible unitary representations of the universal covering group of $ {\text{SL}}(2,{\mathbf{R}})$, Mem. Amer. Math. Soc. No. 69 (1967), 94 pp. MR 0235068 (38:3380)
  • [17] G. Schiffman, Intégrales d'entralacement et fonctions de Whittaker, Bull. Soc. Math. France 99 (1971), 3-72.
  • [18] R. J. Stanke, Analytic uniformly bounded representations of $ {\text{SU}}(1, n + 1)$, Trans. Amer. Math. Soc. 290 (1985), 281-302. MR 787966 (86j:22023)
  • [19] F. Treves, Topological vector spaces, distributions and kernels, 1st ed., Academic Press, New York, 1967. MR 0225131 (37:726)
  • [20] N. R. Wallach, Harmonic analysis on semisimple Lie groups. I, 1st ed., Dekker, New York, 1973. MR 0498996 (58:16978)
  • [21] D. P. Zelobenko, Discrete symmetry operators for reductive Lie groups, Izv. Akad. Nauk SSSR Ser. Mat. 40 (1976), 1055-1083; English transl., Math. USSR-Izv. 10 (1976). MR 0430166 (55:3173)
  • [22] -, A description of the quasi-simple irreducible representations of the groups $ {\text{U}}\,(n,1)$ and $ {\text{Spin}}\,(n,1)$, Izv. Akad. Mat. 41 (1977), 31-50; English transl., Math. USSR Izv. (1977).

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DOI: https://doi.org/10.1090/S0002-9947-1991-1043858-7
Article copyright: © Copyright 1991 American Mathematical Society

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