Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

An orthogonal family of polynomials
on the generalized unit disk
and ladder representations of $U(p,q)$


Author: John D. Lorch
Journal: Proc. Amer. Math. Soc. 126 (1998), 3755-3762
MSC (1991): Primary 22E45, 22E70; Secondary 32L25, 32M15, 58G05, 81R05, 81R25
DOI: https://doi.org/10.1090/S0002-9939-98-04506-7
MathSciNet review: 1458255
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Inner product structures are given for realizations of the positive spin ladder representations over the generalized unit disk ${\bf D}_{p,q} =U(p,q)/K$. This is accomplished by combining previous results of the author with the construction of a family of holomorphic polynomials on ${\bf D}_{p,q}$. These polynomials, which play a crucial role in the present work, are shown to be orthogonal with respect to Lebesgue measure, and their norms are computed. The orthogonal family is then used to invert a certain integral transform, giving the desired inner product structures.


References [Enhancements On Off] (What's this?)

  • 1. M. Atiyah and W. Schmid, A geometric construction of the discrete series for semisimple Lie groups, Invent. Math. 42 (1977), 1-62. MR 57:3310
  • 2. V. Bargmann, On a Hilbert space of analytic functions and an associated integral transform Part I, J. Funct. Anal. 14 (1961), 187-214. MR 28:486
  • 3. T. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, 1978. MR 58:1979
  • 4. M. Davidson, The harmonic representation of $U(p,q)$ and its connection with the generalized unit disk, Pacific J. Math. 129 (1987), 33-55. MR 89c:22021
  • 5. M. Davidson, T. Enright, and R. Stanke, Differential operators and highest weight representations, Mem. Amer. Math. Soc. 455 (1991), 1-117. MR 92c:22034
  • 6. J. Faraut and A. Koranyi, Function spaces and reproducing kernels on bounded symmetric domains, J. Funct. Anal. 88 (1990), 64-89. MR 90m:32049
  • 7. Harish-Chandra, Representations of semisimple Lie groups V, Amer. J. Math. 78 (1956), 1-41. MR 18:490c
  • 8. S. Helgason, Groups and Geometric Analysis, Academic Press, Orlando, 1984. MR 86c:22017
  • 9. L. K. Hua, Harmonic Analysis of Functions of Several Complex Variables, American Mathematical Society, Providence, 1963. MR 30:2162
  • 10. J. Lorch, Unitary structures for ladder representations of $U(p,q)$, Ph.D. thesis, Oklahoma State University, 1995.
  • 11. J. Lorch and L. Mantini, Inversion of an integral transform and ladder representations of $U(1,q)$, Representation Theory and Harmonic Analysis: A Conference in Honor of Ray Kunze (Tuong Ton-That, ed.), Comtemp. Math. 191 (1995), 117-137. MR 96m:22022
  • 12. J. Lorch, An integral transform and ladder representations of $U(p,q)$, Pacific J. Math., in press.
  • 13. L. Mantini, An integral transform in $L^2$-cohomology for the ladder representations of $U(p,q)$, J. Funct. Anal. 60 (1985), 211-241. MR 87a:22029
  • 14. L. Mantini, An $L^2$-cohomology construction of negative spin mass zero equations for $U(p,q)$, J. Math. Anal. Appl. 136 (1988), 419-449. MR 90c:22040
  • 15. M. Narasimhan and K. Okamoto, An analog of the Borel-Weil-Bott theorem for Hermitian symmetric pairs of non-compact type, Ann. of Math. 91 (1970), 486-511. MR 43:419
  • 16. R. Parthasarathy, Dirac operator and the discrete series, Ann. of Math. 96 (1972), 1-30. MR 47:6945
  • 17. I.I. Piatetskii-Shapiro, Automorphic Functions and the Geometry of the Classical Domains, Gordon and Breach, New York, 1969. MR 40:5908
  • 18. W. Rudin, Function Theory in the Unit Ball of ${\bf C}^n$, Springer-Verlag, New York, 1980. MR 82i:32002
  • 19. G. Szegö, Orthogonal Polynomials, American Mathematical Society, Providence, 1939. MR 1:14b

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 22E45, 22E70, 32L25, 32M15, 58G05, 81R05, 81R25

Retrieve articles in all journals with MSC (1991): 22E45, 22E70, 32L25, 32M15, 58G05, 81R05, 81R25


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-98-04506-7
Keywords: Ladder representations, unitary structures, Penrose transform, generalized unit disk
Received by editor(s): January 2, 1997
Received by editor(s) in revised form: April 28, 1997
Communicated by: Roe Goodman
Article copyright: © Copyright 1998 American Mathematical Society

American Mathematical Society