Homogeneous distributions on the Heisenberg group and representations of 𝑆𝑈(2,1)

Fabec, R. C.

doi:10.1090/S0002-9947-1991-1043858-7

Homogeneous distributions on the Heisenberg group and representations of $\textrm {SU}(2,1)$
HTML articles powered by AMS MathViewer

by R. C. Fabec PDF

Trans. Amer. Math. Soc. 328 (1991), 351-391 Request permission

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

Handbook of mathematical functions

L. Corwin and F. P. Greenleaf, Fourier transforms of smooth functions on certain nilpotent Lie groups, J. Functional Analysis 37 (1980), no. 2, 203–217. MR 578932, DOI 10.1016/0022-1236(80)90041-5
Michael Cowling and Adam Korányi, Harmonic analysis on Heisenberg type groups from a geometric viewpoint, Lie group representations, III (College Park, Md., 1982/1983) Lecture Notes in Math., vol. 1077, Springer, Berlin, 1984, pp. 60–100. MR 765552, DOI 10.1007/BFb0072337
Michael Cowling, Harmonic analysis on some nilpotent Lie groups (with application to the representation theory of some semisimple Lie groups), Topics in modern harmonic analysis, Vol. I, II (Turin/Milan, 1982) Ist. Naz. Alta Mat. Francesco Severi, Rome, 1983, pp. 81–123. MR 748862
Raymond C. Fabec, Localizable representations of the de Sitter group, J. Analyse Math. 35 (1979), 151–208. MR 555303, DOI 10.1007/BF02791065
Roger Howe, Quantum mechanics and partial differential equations, J. Functional Analysis 38 (1980), no. 2, 188–254. MR 587908, DOI 10.1016/0022-1236(80)90064-6
A. W. Knapp and E. M. Stein, Intertwining operators for semisimple groups, Ann. of Math. (2) 93 (1971), 489–578. MR 460543, DOI 10.2307/1970887
Hrvoje Kraljević, Representations of the universal convering group of the group $\textrm {SU}(n,\,1)$, Glasnik Mat. Ser. III 8(28) (1973), 23–72 (English, with Serbo-Croatian summary). MR 330355
Hrvoje Kraljević, On representations of the group $SU(n,1)$, Trans. Amer. Math. Soc. 221 (1976), no. 2, 433–448. MR 409725, DOI 10.1090/S0002-9947-1976-0409725-6
R. A. Kunze and E. M. Stein, Uniformly bounded representations and harmonic analysis of the $2\times 2$ real unimodular group, Amer. J. Math. 82 (1960), 1–62. MR 163988, DOI 10.2307/2372876
R. A. Kunze and E. M. Stein, 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 163989, DOI 10.2307/2372907
R. A. Kunze and E. M. Stein, Uniformly bounded representations. III. Intertwining operators for the principal series on semisimple groups, Amer. J. Math. 89 (1967), 385–442. MR 231943, DOI 10.2307/2373128
Ronald L. Lipsman, Uniformly bounded representations of the Lorentz groups, Amer. J. Math. 91 (1969), 938–962. MR 267044, DOI 10.2307/2373311
George W. Mackey, Unitary representations of group extensions. I, Acta Math. 99 (1958), 265–311. MR 98328, DOI 10.1007/BF02392428
Calvin C. Moore and Joseph A. Wolf, Square integrable representations of nilpotent groups, Trans. Amer. Math. Soc. 185 (1973), 445–462 (1974). MR 338267, DOI 10.1090/S0002-9947-1973-0338267-9
Paul J. Sally Jr., Analytic continuation of the irreducible unitary representations of the universal covering group of $\textrm {SL}(2,\,R)$, Memoirs of the American Mathematical Society, No. 69, American Mathematical Society, Providence, R.I., 1967. MR 0235068

Intégrales d’entralacement et fonctions de Whittaker

99

Ronald J. Stanke, Analytic uniformly bounded representations of $\textrm {SU}(1,n+1)$, Trans. Amer. Math. Soc. 290 (1985), no. 1, 281–302. MR 787966, DOI 10.1090/S0002-9947-1985-0787966-1
François Trèves, Topological vector spaces, distributions and kernels, Academic Press, New York-London, 1967. MR 0225131
Nolan R. Wallach, Harmonic analysis on homogeneous spaces, Pure and Applied Mathematics, No. 19, Marcel Dekker, Inc., New York, 1973. MR 0498996
D. P. Želobenko, Discrete symmetry operators for reductive Lie groups, Izv. Akad. Nauk SSSR Ser. Mat. 40 (1976), no. 5, 1055–1083, 1199 (Russian). MR 0430166

A description of the quasi-simple irreducible representations of the groups

and

41