Tensor Products of Analytic Continuations of Holomorphic Discrete Series

Ørsted, Bent ; Zhang, Genkai

Canadian journal of mathematics, Tome 49 (1997) no. 6, pp. 1224-1241

Résumé

We give the irreducible decomposition of the tensor product of an analytic continuation of the holomorphic discrete series of SU(2, 2) with its conjugate.

Détail

DOI : 10.4153/CJM-1997-060-5

Mots-clés : 22E45, 43A85, 32M15, 33A65, Holomorphic discrete series, tensor product, spherical function, Clebsch-Gordan coefficient, Plancherel theorem

Bibliographie

Askey, R. and Wilson, J., Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials. Mem. Amer. Math. Soc., (319) 54(1985). Google Scholar

Berezin, F.A., General concept of quantization. Commun. Math. Phys. 40(1975. 153–174. Google Scholar

Enright, T.J., Howe, R. and Wallach, N.R., A classification of unitary highest weight modules. Progr. Math. 40(1983), 97–143. Google Scholar

Faraut, J. and Koranyi, A., Function spaces and reproducing kernels on bounded symmetric domains. J. Funct. Anal. 89(1990), 64–89. Google Scholar

Helgason, S., Groups and geometric analysis. Academic Press, London, 1984. Google Scholar

Hoogenboom, B., Spherical functions and differential operators on complex Grassmann manifolds. Ark. Mat. 20(1982), 69–85. Google Scholar

Howe, R.,On Some Results of Strichartz and of Rallis and Schiffman. J. Funct. Anal. 32(1979), 297–303. Google Scholar

Hua, L.K., Harmonic analysis of functions of several complex variables in the classical domains. Amer. Math. Soc., Providence, Rhode Island, 1963. Google Scholar

Karlin, S. and McGregor, J., Some properties of orthogonal polynomials. In: Theory and Application of Special Functions (Ed. Askey, R.), Academic Press, 1975. 521–550. Google Scholar

Knapp, A.W., Representation Theory of Semisimple Groups. Princeton University Press, Princeton, New Jersey, 1986. Google Scholar

Knapp, A.W. and Speh, B., Irreducible Unitary Representations of SU(2, 2). J. Funct. Anal. 45(1982), 41–73. Google Scholar

Magnus, W. and Oberhettinger, F., Formulas and theorems for the functions of mathematical physics. Chelsea, New York, 1949. Google Scholar

Mautner, F.I., Unitary representations of locally compact groups. II. Ann. of Math. 52(1950), 528–556. Google Scholar

Opdam, E., Private communication. Google Scholar

Ørsted, B., Composition series for analytic continuations of holomorphic discrete series representations of SU(n, n ). Trans. Amer.Math. Soc. 260(1980), 563–573. Google Scholar

Ørsted, B. and Zhang, G., L2-versions of the Howe Correspondence I. Math. Scand., to appear. Google Scholar

Peetre, J. and Zhang, G., A weighted Plancherel formula III. The case of a hyperbolic matrix ball. Collect. Math. 43(1992), 273–301. Google Scholar

Pukánszky, L., On the Kronecker products of irreducible representations of the 2 ð 2 real unimodular group. Part I. Trans. Amer.Math. Soc. 100(1961), 116–152. Google Scholar

Repka, J., Tensor products of unitary representations of SL2(R).Amer. J. Math. 100(1978), 930–932. Google Scholar

Repka, J., Tensor products of holomorphic discrete series. Can. J. Math. 31(1979), 836–844. Google Scholar

Segal, I.E., An extension of Plancherel's formula to separable unimodular groups. Ann. of Math. 52(1950), 272–292. Google Scholar

Unterberger, A. and Upmeier, H., Berezin transform and invariant differential operators. Preprint. Google Scholar

Wallach, N.R., The analytic continuation of the discrete series I, II. Trans. Amer.Math. Soc. 251(1979), 1–17. 19–37. Google Scholar

Wilson, J., Some hypergeometric orthogonal polynomials. SIAM. J. Math. Anal. 11(1980), 690–701. Google Scholar

Zhang, G., Tensor products of weighted Bergman spaces and invariant Ha-plitz operators. Math. Scand. 71(1992), 85–95. Google Scholar