Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Some recurrence formulas for spherical polynomials on tube domains


Author: Gen Kai Zhang
Journal: Trans. Amer. Math. Soc. 347 (1995), 1725-1734
MSC: Primary 22E46; Secondary 33C55, 65D15
DOI: https://doi.org/10.1090/S0002-9947-1995-1249896-7
MathSciNet review: 1249896
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For a tube domain $ G/K$ we study the tensor products of two spherical representations of the maximal compact group $ K$ and the product of the corresponding spherical polynomials. When one of these is a fundamental representation, we prove that the spherical representations appear with multiplicity at most one and we then find all the coefficients in the recurrence formula for the product of the spherical polynomials. This generalizes the previous result of L. Vretare and proves for certain cases a conjecture of R. Stanley on Jack symmetric polynomials.


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

  • [1] N. Bourbaki, Groupes et algebres de Lie, Chapitres 4, 5 et 6, Hermann, Paris, 1975. MR 0453824 (56:12077)
  • [2] -, Groupes et algebres de Lie, Chapitres 7 et 8, Hermann, Paris, 1975.
  • [3] T. J. Enright and A. Joseph, An intrinsic analysis of unitarizabel highest weight modules, Math. Ann. 288 (1990), 571-594. MR 1081264 (91m:17005)
  • [4] J. Faraut and A. Koranyi, Function spaces and reproducing kernels on bounded symmetric, J. Funct. Anal. 89 (1990), 64-89. MR 1033914 (90m:32049)
  • [5] S. Helgason, Differential geometry, Lie groups and symmetric spaces, Academic Press, London, 1978. MR 514561 (80k:53081)
  • [6] -, Groups and geometric analysis, Academic Press, London, 1984. MR 754767 (86c:22017)
  • [7] B. Kostant and S. Sahi, The Capelli identity, tube domains, and the generalized Laplace transform, Adv. Math. 87 (1991), 71-92. MR 1102965 (92h:22033)
  • [8] O. Loos, Bounded symmetric domains and Jordan pairs, University of California, Irvine, 1977.
  • [9] I. G. Macdonald, Symmetric functions and Hall polynomials, Oxford University Press, New York, 1979. MR 553598 (84g:05003)
  • [10] -, Commuting differential operators and zonal spherical functions, Algebraic groups, Utrecht 1986, Lectures Notes in Math., vol. 1271, Springer-Verlag, New York, 1987. MR 911140 (89e:43025)
  • [11] B. Ørsted and G. Zhang, Reproducing kernels and composition series for spaces of vector-valued holomorphic functions on tube domains, J. Funct. Anal. 124 (1994), 181-204. MR 1284609 (96a:32060)
  • [12] W. Schmid, Die Randwerte holomorpher Functionen auf hermitesch symmetrischen Räumen, Invent. Math. 9 (1969), 61-80. MR 0259164 (41:3806)
  • [13] R. P. Stanley, Some combinatorial properties of Jack symmetric functions, Adv. Math. 77 (1989), 76-115. MR 1014073 (90g:05020)
  • [14] H. Upmeier, Jordan algebras and harmonic analysis on symmetric spaces, Amer. J. Math. 108 (1986), 1-25. MR 821311 (87e:32047)
  • [15] -, Toeplitz operators on bounded symmetric domains, Tran. Amer. Math. Soc. 280 (1983), 221-237. MR 712257 (85g:47042)
  • [16] L. Vretare, Elementary spherical functions on symmetric spaces, Math. Scand. 39 (1976), 343-358. MR 0447979 (56:6289)
  • [17] Z. Yan, Thesis, CUNY Graduate School, 1990.
  • [18] D. P. Zhelobenko, Compact Lie groups and their representations, Transl. Math. Monographs, vol. 40, Amer. Math. Soc., Providence, RI, 1973.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 22E46, 33C55, 65D15

Retrieve articles in all journals with MSC: 22E46, 33C55, 65D15


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1995-1249896-7
Keywords: Tube domain, recurrence formula, spherical polynomial
Article copyright: © Copyright 1995 American Mathematical Society

American Mathematical Society