We study the relative discrete series of the -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.
Cet article est une étude de la série relative discrète de l’espace des sections 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.
@article{AIF_1996__46_4_1011_0, 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}, number = {4}, year = {1996}, doi = {10.5802/aif.1538}, mrnumber = {98b:22028}, zbl = {0853.22011}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/aif.1538/} }
TY - JOUR AU - Dooley, Anthony H. AU - Ørsted, Bent AU - Zhang, Genkai TI - Relative discrete series of line bundles over bounded symmetric domains JO - Annales de l'Institut Fourier PY - 1996 SP - 1011 EP - 1026 VL - 46 IS - 4 PB - Association des Annales de l’institut Fourier UR - http://archive.numdam.org/articles/10.5802/aif.1538/ DO - 10.5802/aif.1538 LA - en ID - AIF_1996__46_4_1011_0 ER -
%0 Journal Article %A Dooley, Anthony H. %A Ørsted, Bent %A Zhang, Genkai %T Relative discrete series of line bundles over bounded symmetric domains %J Annales de l'Institut Fourier %D 1996 %P 1011-1026 %V 46 %N 4 %I Association des Annales de l’institut Fourier %U http://archive.numdam.org/articles/10.5802/aif.1538/ %R 10.5802/aif.1538 %G en %F AIF_1996__46_4_1011_0
Dooley, Anthony H.; Ørsted, Bent; Zhang, Genkai. Relative discrete series of line bundles over bounded symmetric domains. Annales de l'Institut Fourier, Volume 46 (1996) no. 4, pp. 1011-1026. doi : 10.5802/aif.1538. http://archive.numdam.org/articles/10.5802/aif.1538/
[CM] Asymptotic behaviour of matrix coefficients of admissible representations, Duke Math. J., 49 (1982), 869-930. | MR | Zbl
& ,[FK] Function spaces and reproducing kernels on bounded symmetric domains, J. Funct. Anal., 89 (1990), 64-89. | MR | Zbl
& ,[HC] Representations of semisimple Lie groups IV, Amer. J. Math., 77 (1955), 743-777. | MR | Zbl
,[He] Groups and geometric analysis, Academic Press, London, 1984. | Zbl
,[JP] A new generalization of Hankel operators, Math. Nachr., 132 (1987), 313-328. | MR | Zbl
& ,[J] On a ring of invariant polynomials on a Hermitian symmetric space, J. of Algebra, 67 (1980), 72-81. | MR | Zbl
,[LP] Weighted Plancherel formula. Irreducible unitary representations and eigenspace representations, Math. Scand., 72 (1993), 99-119. | Zbl
& ,[L] Bounded Symmetric Domains and Jordan Pairs, University of California, Irvine, 1977. | Zbl
,[OZ1] Tensor products of analytic continuations of holomorphic discrete series, preprint, 1994.
& ,[OZ2]
& , in preparation.[P] Covariant Laplaceans and Cauchy-Riemann operators for a Cartan domain, manuscript.
,[PPZ] A weighted Plancherel formula I. The case of the unit disk. Application to Hankel operators, technical report, Stockholm, 1990.
, & ,[PZ] A weighted Plancherel formula III. The case of a hyperbolic matrix ball, Collect. Math., 43 (1992), 273-301. | Zbl
& ,[Re] Tensor products of holomorphic discrete series, Can. J. Math., 31 (1979), 836-844. | MR | Zbl
,[Sa] Algebraic structures of symmetric domains, Iwanami Shoten and Princeton Univ. Press, Tokyo and Princeton, NJ, 1980. | MR | Zbl
,[Sch] 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] 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] Jordan algebras and harmonic analysis on symmetric spaces, Amer. J. Math., 108 (1986), 1-25. | MR | Zbl
,[W] The analytic continuation of the discrete series. I, II, Trans. Amer. Math. Soc., 251 (1979), 1-17; 19-37. | MR | Zbl
,[Ze] Compact Lie groups and their representations, Amer. Math. Soc., Transl. Math. Monographs, vol. 40, Providence, Rhode Island, 1973. | Zbl
,[Zh1] Ha-plitz operators between Moebius invariant subspaces, Math. Scand., 71 (1992), 69-84. | MR | Zbl
,[Zh2] A weighted Plancherel formula II. The case of the ball, Studia Math., 102 (1992), 103-120. | MR | Zbl
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