Combinatorics and spherical functions on the Heisenberg group
Authors:
Chal Benson and Gail Ratcliff
Journal:
Represent. Theory 2 (1998), 79105
MSC (1991):
Primary 22E30, 43A55
Posted:
April 1, 1998
MathSciNet review:
1613063
Fulltext PDF Free Access
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Abstract: Let be a finite dimensional Hermitian vector space and be a compact Lie subgroup of for which the representation of on is multiplicity free. One obtains a canonical basis for the space of invariant polynomials on and also a basis via orthogonalization of the 's. The polynomial yields the homogeneous component of highest degree in . The coefficients that express the 's in terms of the 's are the generalized binomial coefficients of Z. Yan. We present some new combinatorial identities that involve these coefficients. These have applications to analysis on Heisenberg groups. Indeed, the polynomials completely determine the generic bounded spherical functions for a Gelfand pair obtained from the action of on a Heisenberg group .
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 C. Benson, J. Jenkins, and G. Ratcliff, spherical functions on Heisenberg groups. Contemporary Math. 145, 181197, 1993. MR 94h:22002
 [BJRW96]
 C. Benson, J. Jenkins, G. Ratcliff, and T. Worku, Spectra for Gelfand pairs associated with the Heisenberg group. Colloq. Math. 71, 305328, 1996. MR 98b:22016
 [BR96]
 C. Benson and G. Ratcliff, A classification for multiplicity free actions. Journal of Algebra 181, 152186, 1996. MR 97c:14046
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 G. Carcano, A commutativity condition for algebras of invariant functions. Boll. Un. Mat. Italiano 7, 10911105, 1987. MR 89h:22011
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 H. Dib, Fonctions de Bessel sur une algèbre de Jordan. J. Math Pures et Appl. 69, 403448, 1990. MR 92e:33017
 [FK94]
 J. Faraut and A. Koranyi, Analysis on Symmetric Cones. Oxford University Press, New York, 1994. CMP 97:12
 [How95]
 R. Howe, Perspectives on invariant theory: Schur duality, multiplicityfree actions and beyond. Israel Math. Conf. Proc., vol. 8, BarIlan Univ., Ramat Gan, 1995. MR 96e:13006
 [HU91]
 R. Howe and T. Umeda, The Capelli identity, the double commutant theorem and multiplicityfree actions. Math. Annalen 290, 565619, 1991. MR 92j:17004
 [Kac80]
 V. Kac, Some remarks on nilpotent orbits. Journal of Algebra 64, 190213, 1980. MR 81i:17005
 [KS96]
 F. Knop and S. Sahi, Difference equations and symmetric polynomials defined by their zeroes. International Math. Research Notes 10, 473486, 1996. CMP 96:15
 [Lea]
 A. Leahy, Ph.D. Thesis, Rutgers University.
 [Mac95]
 I. G. Macdonald, Symmetric Functions and Hall Polynomials, Second Edition. Clarendon Press, Oxford, 1995. MR 96h:05207
 [Ols]
 G. Olshanski, Quasisymmetric functions and factorial Schur functions. (preprint).
 [OO97]
 A. Okounkov and G. Olshanski, Shifted Jack polynomials, binomial formula, and applications. Math. Res. Letters 4, 6978, 1997. CMP 97:08
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 S. Sahi, The spectrum of certain invariant differential operators associated to Hermitian symmetric spaces. Lie Theory and Geometry, vol. 123, Progress in Math.
(J. L. Brylinski, editor), Birkhäuser, Boston, pp. 569576, 1994. MR 96d:43013
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Additional Information
Chal Benson
Affiliation:
Department of Mathematics and Computer Science, University of MissouriSt. Louis, St. Louis, Missouri 63121
Email:
benson@arch.umsl.edu
Gail Ratcliff
Affiliation:
Department of Mathematics and Computer Science, University of MissouriSt. Louis, St. Louis, Missouri 63121
Email:
ratcliff@arch.umsl.edu
DOI:
http://dx.doi.org/10.1090/S1088416598000405
PII:
S 10884165(98)000405
Received by editor(s):
October 22, 1997
Received by editor(s) in revised form:
February 17, 1998
Posted:
April 1, 1998
Article copyright:
© Copyright 1998 American Mathematical Society
