Relative discrete series of line bundles over bounded symmetric domains
Dooley, Anthony H. ; Ørsted, Bent ; Zhang, Genkai
Annales de l'Institut Fourier, Tome 46 (1996), p. 1011-1026 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

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.

@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},
     volume = {46},
     year = {1996},
     pages = {1011-1026},
     doi = {10.5802/aif.1538},
     mrnumber = {98b:22028},
     zbl = {0853.22011},
     language = {en},
     url = {http://dml.mathdoc.fr/item/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, Tome 46 (1996) pp. 1011-1026. doi : 10.5802/aif.1538. http://gdmltest.u-ga.fr/item/AIF_1996__46_4_1011_0/

[CM] W. Casselman & D. Milićic, Asymptotic behaviour of matrix coefficients of admissible representations, Duke Math. J., 49 (1982), 869-930. | MR 85a:22024 | Zbl 0524.22014

[FK] J. Faraut & A. Koranyi, Function spaces and reproducing kernels on bounded symmetric domains, J. Funct. Anal., 89 (1990), 64-89. | MR 90m:32049 | Zbl 0718.32026

[HC] Harish-Chandra, Representations of semisimple Lie groups IV, Amer. J. Math., 77 (1955), 743-777. | MR 17,282c | Zbl 0066.35603

[He] S. Helgason, Groups and geometric analysis, Academic Press, London, 1984. | Zbl 0543.58001

[JP] S. Janson & J. Peetre, A new generalization of Hankel operators, Math. Nachr., 132 (1987), 313-328. | MR 88m:47045 | Zbl 0644.47026

[J] K. D. Johnson, On a ring of invariant polynomials on a Hermitian symmetric space, J. of Algebra, 67 (1980), 72-81. | MR 83c:22019 | Zbl 0491.22007

[LP] H. Liu & L. Peng, Weighted Plancherel formula. Irreducible unitary representations and eigenspace representations, Math. Scand., 72 (1993), 99-119. | Zbl 0785.22018

[L] O. Loos, Bounded Symmetric Domains and Jordan Pairs, University of California, Irvine, 1977. | Zbl 0228.32012

[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 0836.43018

[Re] J. Repka, Tensor products of holomorphic discrete series, Can. J. Math., 31 (1979), 836-844. | MR 82c:22017 | Zbl 0373.22006

[Sa] I. Satake, Algebraic structures of symmetric domains, Iwanami Shoten and Princeton Univ. Press, Tokyo and Princeton, NJ, 1980. | MR 82i:32003 | Zbl 0483.32017

[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 85i:22014 | Zbl 0545.22015

[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 95b:22036 | Zbl 0830.43018

[Up] H. Upmeier, Jordan algebras and harmonic analysis on symmetric spaces, Amer. J. Math., 108 (1986), 1-25. | MR 87e:32047 | Zbl 0603.46055

[W] N. Wallach, The analytic continuation of the discrete series. I, II, Trans. Amer. Math. Soc., 251 (1979), 1-17; 19-37. | MR 81a:22009 | Zbl 0419.22017

[Ze] D. P. Zelobenko, Compact Lie groups and their representations, Amer. Math. Soc., Transl. Math. Monographs, vol. 40, Providence, Rhode Island, 1973. | Zbl 0272.22006

[Zh1] G. Zhang, Ha-plitz operators between Moebius invariant subspaces, Math. Scand., 71 (1992), 69-84. | MR 94e:47039a | Zbl 0793.47028

[Zh2] G. Zhang, A weighted Plancherel formula II. The case of the ball, Studia Math., 102 (1992), 103-120. | MR 94e:43009 | Zbl 0811.43003