Representation Theory

Author(s): Genkai Zhang
Journal: Represent. Theory 5 (2001), 164-190.
MSC (2000): Primary 22E46, 47A70, 32M15, 33C52
Posted: June 15, 2001
Retrieve article in: PDF
This article is available free of charge

Let

be an irreducible bounded symmetric domain with genus

and $H^{\nu}(D)$ the weighted Bergman spaces of holomorphic functions for $\nu >p-1$ . The spaces $H^\nu(D)$ form unitary (projective) representations of the group

and have analytic continuation in $\nu$ ; they give also unitary representations when $\nu$ in the Wallach set, which consists of a continuous part and a discrete part of

points. The first non-trivial discrete point $\nu=\frac a2$ gives the minimal highest weight representation of

. We give the irreducible decomposition of tensor product $H^{\frac a2}\otimes \overline{H^{\frac a2}}$ . As a consequence we discover some new spherical unitary representations of

and find the expansion of the corresponding spherical functions in terms of the

-invariant (Jack symmetric) polynomials, the coefficients being continuous dual Hahn polynomials.

1.: R. Askey and J. Wilson, Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, vol. 54, Mem. Amer. Math. Soc., no. 319, Amer. Math. Soc., 1985. MR 87a:05023
2.: M. G. Davidson, T. J. Enright, and R. J. Stanke, Differential operators and highest weight representations, vol. 94, Memoirs of Amer. Math. Soc., no. 455, Amer. Math. Soc., Providence, Rhode Island, 1991. MR 92c:22034
3.: A. Dvorsky and S. Sahi, Tensor products of singular representations and an extension of the $\theta$ -correspondence, Selecta. Math. 4 (1998), 11-29. MR 99e:22028
4.: M. Englis, S. C. Hille, J. Peetre, H. Rosengren, and G. Zhang, A new kind of Hankel type operators connected with the complementary series, Arabic J. Math. Sci. 6 (2000), 49-80. CMP 2001:06
5.: J. Faraut and A. Koranyi, Function spaces and reproducing kernels on bounded symmetric domains, J. Funct. Anal. 88 (1990), 64-89. MR 90m:32049
6.: G. Gasper and M. Rahman, Basic hypergeometric series, Encyclopedia of Math. and Appl., 35, Cambridge University Press, Cambridge, 1990. MR 91d:33034
7.: S. Helgason, Groups and geometric analysis, Academic Press, New York, London, 1984. MR 86c:22017
8.: S. C. Hille, Canonical representations, Ph.D. thesis, Leiden University, 1999.
9.: B. Hoogenboom, Spherical functions and invariant differential operators on complex Grassmann manifolds, Ark. Mat. 20 (1982), 69-85. MR 83m:43012
10.: L. K. Hua, Harmonic analysis of functions of several complex variables in the classical domains, Amer. Math. Soc., Providence, Rhode Island, 1963. MR 83c:32032
11.: M. Kashiwara and M. Vergne, On the Segal-Shale-Weil representation and harmonic polynomials, Invent. Math. 44 (1978), 1-47. MR 57:3311
12.: A. Kirillov, Merits and demerits of the orbit method, Bull. Amer. Math. Soc. 36 (1999), no. 4, 433-488. MR 2000h:22001
13.: A. Knapp and B. Speh, The role of basic cases in classification: Theorems about unitary representations applicable to , Noncommutative Harmonic Analysis and Lie Groups (Berlin Heidelberg New York) (J. Carmona and M. Vergne, eds.), Lecture Notes in Mathematics, vol. 1020, Springer-Verlag, 1983, pp. 119-160. MR 85i:22026
14.: R. Koekoek and R. F. Swarttouw, The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue, Math. report, Delft Univ. of Technology 98-17, 1998.
15.: I. G. Macdonald, Symmetric functions and Hall polynomials, Second ed., Clarendon Press, Oxford, 1995. MR 96h:05207
16.: E. Nelson and W. F. Stinespring, Representation of elliptic operators in an enveloping algebra, Amer. J. Math. 81 (1959), 547-560. MR 22:907
17.: Yu. Neretin, Plancherel formula for Berezin deformation of ${L^2}$ on Riemannian symmetric space, (1999), preprint, Math.RT/9911020.
18.: G. Ólafsson and B. Ørsted, Generalizations of the Bargmann transform, Lie theory and its applications in physics. Proceedings of the international workshop, Clausthal, Germany, August 14-17, 1995. (H.-D.Doebner et al, ed.), World Scientific, Singapore, 1996, pp. 3-14. MR 99e:22032
19.: B. Ørsted and G. Zhang, Weyl quantization and tensor products of Fock and Bergman spaces, Indiana Univ. Math. J. 43 (1994), 551-583. MR 95h:22008
20.: -, -versions of the Howe correspondence. I, Math. Scand. 80 (1997), 125-160. MR 99c:22017
21.: -, Tensor products of analytic continuations of holomorphic discrete series, Canadian J. Math. 49 (1997), 1224-1241. MR 2000b:22012
22.: J. Peetre and G. Zhang, A weighted Plancherel formula. III, The case of a hyperbolic matrix domain, Collect. Math. 43 (1992), 273-301. MR 94m:43008
23.: J. Repka, Tensor products of unitary representations of ${SL_2(\bf R)}$ , Amer. J. Math. 100 (1978), 747-774. MR 80g:22014
24.: -, Tensor products of holomorphic discrete series representations, Can. J. Math. 31 (1979), 836-844. MR 82c:22017
25.: H. Rossi and M. Vergne, Analytic continuation of the holomorphic discrete series of a semisimple lie group, Acta Math. 136 (1976), 1-59. MR 58:1032
26.: W. Schmid, Die Randwerte holomorpher Funktionen auf hermitesch symmetrischen Räumen, Invent. Math 9 (1969), 61-80. MR 41:3806
27.: G. Shimura, Invariant differential operators on Hermitian symmetric spaces, Ann. Math. 132 (1990), 232-272. MR 91i:22015
28.: -, Differential operators, holomorphic projection, and singular forms, Duke. Math. J. 76 (1994), no. 1, 141-173. MR 95k:11072
29.: H. Upmeier, Toeplitz operators on bounded symmetric domains, Trans. Amer. Math. Soc. 280 (1983), 221-237. MR 85g:47042
30.: N. Wallach, The analytic continuation of the discrete series, I, II, Trans. Amer. Math. Soc. 251 (1979), 1-17; 19-37. MR 81a:22009
31.: J. Wilson, Some hypergeometric orthogonal polynomials, SIAM. J. Math. Anal. 11 (1980), 690-701. MR 82a:33014
32.: Z. Yan, Differential operators and function spaces, Several complex variables in China, Contemp. Math., vol. 142, Amer. Math. Soc., 1993, pp. 121-142. MR 93m:32047
33.: G. Zhang, Tensor products of weighted Bergman spaces and invariant Ha-plitz operators, Math. Scand. 71 (1992), 85-95. MR 94e:47039b
34.: -, A weighted Plancherel formula II. The case of a ball, Studia Math. 102 (1992), 103-120. MR 94e:43009
35.: -, Invariant differential operators on hermitian symmetric spaces and their eigenvalues, Israel J. Math. 119 (2000), 157-185. CMP 2001:06
36.: -, Shimura invariant differential operators and their eigenvalues, Math. Ann., 319 (2001), 235-265. CMP 2001:09
37.: -, Berezin transform on line bundles over bounded symmetric domains, J. Lie Theory 10 (2000), 111-126. MR 2001c:32015

Retrieve articles in Representation Theory with MSC (2000): 22E46, 47A70, 32M15, 33C52

Genkai Zhang
Affiliation: Department of Mathematics, Chalmers University of Technology and Göteborg University, S-412 96 Göteborg, Sweden
Email: genkai@math.chalmers.se

DOI: 10.1090/S1088-4165-01-00103-0
PII: S 1088-4165(01)00103-0
Keywords: Bounded symmetric domains, weighted Bergman spaces, unitary highest weight representations, invariant differential operators, tensor product, irreducible decomposition, Clebsch-Gordan coefficients
Received by editor(s): May 23, 2000
Received by editor(s) in revised form: April 10, 2001
Posted: June 15, 2001
Additional Notes: Research supported by the Swedish Natural Science Research Council (NFR)
Copyright of article: Copyright 2001, American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-4165

Previous volume \| Table of contents \| Next volume Articles in press \| Most recent volume \| All volumes Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2009, American Mathematical Society Privacy Statement		Search the AMS