no. 4
Relative discrete series of line bundles over bounded symmetric domains
Annales de l'Institut Fourier, Tome 46 (1996) no. 4, pp. 1011-1026.

Cet article est une étude de la série relative discrète de l’espace des sections L 2 d’un fibré sur un domaine symétrique borné. On démontre que toute série discrète provient, en tant que sous-module irréductible, d’un produit tensoriel d’une série holomorphe discrète par une représentation de dimension finie.

We study the relative discrete series of the L 2 -space of the sections of a line bundle over a bounded symmetric domain. We prove that all the discrete series appear as irreducible submodules of the tensor product of a holomorphic discrete series with a finite dimensional representation.

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     author = {Dooley, Anthony H. and {\O}rsted, Bent and Zhang, Genkai},
     title = {Relative discrete series of line bundles over bounded symmetric domains},
     journal = {Annales de l'Institut Fourier},
     pages = {1011--1026},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {46},
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Dooley, Anthony H.; Ørsted, Bent; Zhang, Genkai. Relative discrete series of line bundles over bounded symmetric domains. Annales de l'Institut Fourier, Tome 46 (1996) no. 4, pp. 1011-1026. doi : 10.5802/aif.1538. https://aif.centre-mersenne.org/articles/10.5802/aif.1538/

[CM] W. Casselman & D. Milićic, Asymptotic behaviour of matrix coefficients of admissible representations, Duke Math. J., 49 (1982), 869-930. | MR | Zbl

[FK] J. Faraut & A. Koranyi, Function spaces and reproducing kernels on bounded symmetric domains, J. Funct. Anal., 89 (1990), 64-89. | MR | Zbl

[HC] Harish-Chandra, Representations of semisimple Lie groups IV, Amer. J. Math., 77 (1955), 743-777. | MR | Zbl

[He] S. Helgason, Groups and geometric analysis, Academic Press, London, 1984. | Zbl

[JP] S. Janson & J. Peetre, A new generalization of Hankel operators, Math. Nachr., 132 (1987), 313-328. | MR | Zbl

[J] K. D. Johnson, On a ring of invariant polynomials on a Hermitian symmetric space, J. of Algebra, 67 (1980), 72-81. | MR | Zbl

[LP] H. Liu & L. Peng, Weighted Plancherel formula. Irreducible unitary representations and eigenspace representations, Math. Scand., 72 (1993), 99-119. | Zbl

[L] O. Loos, Bounded Symmetric Domains and Jordan Pairs, University of California, Irvine, 1977. | Zbl

[OZ1] B. Ørsted & G. Zhang, Tensor products of analytic continuations of holomorphic discrete series, preprint, 1994.

[OZ2] B. Ørsted & G. Zhang, in preparation.

[P] J. Peetre, Covariant Laplaceans and Cauchy-Riemann operators for a Cartan domain, manuscript.

[PPZ] J. Peetre, L. Peng & G. Zhang, A weighted Plancherel formula I. The case of the unit disk. Application to Hankel operators, technical report, Stockholm, 1990.

[PZ] J. Peetre & G. Zhang, A weighted Plancherel formula III. The case of a hyperbolic matrix ball, Collect. Math., 43 (1992), 273-301. | Zbl

[Re] J. Repka, Tensor products of holomorphic discrete series, Can. J. Math., 31 (1979), 836-844. | MR | Zbl

[Sa] I. Satake, Algebraic structures of symmetric domains, Iwanami Shoten and Princeton Univ. Press, Tokyo and Princeton, NJ, 1980. | MR | Zbl

[Sch] H. Schlichtkrull, One-dimensional K-types in finite dimensional representations of semisimple Lie groups : A generalization of Helgason's theorem, Math. Scand., 54 (1984), 279-294. | MR | Zbl

[Sh] N. Shimeno, The Plancherel formula for spherical functions with one-dimensional K-type on a simply connected simple Lie group of Hermitian type, J. Funct. Anal., 121 (1994), 331-388. | MR | Zbl

[Up] H. Upmeier, Jordan algebras and harmonic analysis on symmetric spaces, Amer. J. Math., 108 (1986), 1-25. | MR | Zbl

[W] N. Wallach, The analytic continuation of the discrete series. I, II, Trans. Amer. Math. Soc., 251 (1979), 1-17; 19-37. | MR | Zbl

[Ze] D. P. Zelobenko, Compact Lie groups and their representations, Amer. Math. Soc., Transl. Math. Monographs, vol. 40, Providence, Rhode Island, 1973. | Zbl

[Zh1] G. Zhang, Ha-plitz operators between Moebius invariant subspaces, Math. Scand., 71 (1992), 69-84. | MR | Zbl

[Zh2] G. Zhang, A weighted Plancherel formula II. The case of the ball, Studia Math., 102 (1992), 103-120. | MR | Zbl

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