Relative discrete series of line bundles over bounded symmetric domains

Anthony H. Dooley; Bent Ørsted; Genkai Zhang

Annales de l'institut Fourier (1996)

  • Volume: 46, Issue: 4, page 1011-1026
  • ISSN: 0373-0956

Abstract

top
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.

How to cite

top

Dooley, Anthony H., Ørsted, Bent, and Zhang, Genkai. "Relative discrete series of line bundles over bounded symmetric domains." Annales de l'institut Fourier 46.4 (1996): 1011-1026. <http://eudml.org/doc/75197>.

@article{Dooley1996,
abstract = {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.},
author = {Dooley, Anthony H., Ørsted, Bent, Zhang, Genkai},
journal = {Annales de l'institut Fourier},
keywords = {bounded symmetric domain; line bundle; relative discrete series; holomorphic discrete series},
language = {eng},
number = {4},
pages = {1011-1026},
publisher = {Association des Annales de l'Institut Fourier},
title = {Relative discrete series of line bundles over bounded symmetric domains},
url = {http://eudml.org/doc/75197},
volume = {46},
year = {1996},
}

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
PB - Association des Annales de l'Institut Fourier
VL - 46
IS - 4
SP - 1011
EP - 1026
AB - 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.
LA - eng
KW - bounded symmetric domain; line bundle; relative discrete series; holomorphic discrete series
UR - http://eudml.org/doc/75197
ER -

References

top
  1. [CM] W. CASSELMAN & D. MILIĆIC, Asymptotic behaviour of matrix coefficients of admissible representations, Duke Math. J., 49 (1982), 869-930. Zbl0524.22014MR85a:22024
  2. [FK] J. FARAUT & A. KORANYI, Function spaces and reproducing kernels on bounded symmetric domains, J. Funct. Anal., 89 (1990), 64-89. Zbl0718.32026MR90m:32049
  3. [HC] HARISH-CHANDRA, Representations of semisimple Lie groups IV, Amer. J. Math., 77 (1955), 743-777. Zbl0066.35603MR17,282c
  4. [He] S. HELGASON, Groups and geometric analysis, Academic Press, London, 1984. Zbl0543.58001
  5. [JP] S. JANSON & J. PEETRE, A new generalization of Hankel operators, Math. Nachr., 132 (1987), 313-328. Zbl0644.47026MR88m:47045
  6. [J] K. D. JOHNSON, On a ring of invariant polynomials on a Hermitian symmetric space, J. of Algebra, 67 (1980), 72-81. Zbl0491.22007MR83c:22019
  7. [LP] H. LIU & L. PENG, Weighted Plancherel formula. Irreducible unitary representations and eigenspace representations, Math. Scand., 72 (1993), 99-119. Zbl0785.22018
  8. [L] O. LOOS, Bounded Symmetric Domains and Jordan Pairs, University of California, Irvine, 1977. Zbl0228.32012
  9. [OZ1] B. ØRSTED & G. ZHANG, Tensor products of analytic continuations of holomorphic discrete series, preprint, 1994. 
  10. [OZ2] B. ØRSTED & G. ZHANG, in preparation. 
  11. [P] J. PEETRE, Covariant Laplaceans and Cauchy-Riemann operators for a Cartan domain, manuscript. 
  12. [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. 
  13. [PZ] J. PEETRE & G. ZHANG, A weighted Plancherel formula III. The case of a hyperbolic matrix ball, Collect. Math., 43 (1992), 273-301. Zbl0836.43018
  14. [Re] J. REPKA, Tensor products of holomorphic discrete series, Can. J. Math., 31 (1979), 836-844. Zbl0373.22006MR82c:22017
  15. [Sa] I. SATAKE, Algebraic structures of symmetric domains, Iwanami Shoten and Princeton Univ. Press, Tokyo and Princeton, NJ, 1980. Zbl0483.32017MR82i:32003
  16. [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. Zbl0545.22015MR85i:22014
  17. [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. Zbl0830.43018MR95b:22036
  18. [Up] H. UPMEIER, Jordan algebras and harmonic analysis on symmetric spaces, Amer. J. Math., 108 (1986), 1-25. Zbl0603.46055MR87e:32047
  19. [W] N. WALLACH, The analytic continuation of the discrete series. I, II, Trans. Amer. Math. Soc., 251 (1979), 1-17; 19-37. Zbl0419.22017MR81a:22009
  20. [Ze] D. P. ZELOBENKO, Compact Lie groups and their representations, Amer. Math. Soc., Transl. Math. Monographs, vol. 40, Providence, Rhode Island, 1973. Zbl0272.22006
  21. [Zh1] G. ZHANG, Ha-plitz operators between Moebius invariant subspaces, Math. Scand., 71 (1992), 69-84. Zbl0793.47028MR94e:47039a
  22. [Zh2] G. ZHANG, A weighted Plancherel formula II. The case of the ball, Studia Math., 102 (1992), 103-120. Zbl0811.43003MR94e:43009

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.