Combinatorics and spherical functions

on the Heisenberg group

Authors:
Chal Benson and Gail Ratcliff

Journal:
Represent. Theory **2** (1998), 79-105

MSC (1991):
Primary 22E30, 43A55

DOI:
https://doi.org/10.1090/S1088-4165-98-00040-5

Published electronically:
April 1, 1998

MathSciNet review:
1613063

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Abstract | References | Similar Articles | Additional Information

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 .

**[BJR]**C. Benson, J. Jenkins, and G. Ratcliff, The spherical transform of a Schwartz function on the Heisenberg group. (*to appear in J. Functional Analysis*).**[BJR92]**C. Benson, J. Jenkins, and G. Ratcliff, Bounded -spherical functions on Heisenberg groups.*J. Functional Analysis***105**, 409-443, 1992. MR**93e:22017****[BJR93]**C. Benson, J. Jenkins, and G. Ratcliff, -spherical functions on Heisenberg groups.*Contemporary Math.***145**, 181-197, 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**, 305-328, 1996. MR**98b:22016****[BR96]**C. Benson and G. Ratcliff, A classification for multiplicity free actions.*Journal of Algebra***181**, 152-186, 1996. MR**97c:14046****[Car87]**G. Carcano, A commutativity condition for algebras of invariant functions.*Boll. Un. Mat. Italiano***7**, 1091-1105, 1987. MR**89h:22011****[Dib90]**H. Dib, Fonctions de Bessel sur une algèbre de Jordan.*J. Math Pures et Appl.***69**, 403-448, 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, multiplicity-free actions and beyond.**Israel Math. Conf. Proc.*, vol. 8, Bar-Ilan Univ., Ramat Gan, 1995. MR**96e:13006****[HU91]**R. Howe and T. Umeda, The Capelli identity, the double commutant theorem and multiplicity-free actions.*Math. Annalen***290**, 565-619, 1991. MR**92j:17004****[Kac80]**V. Kac, Some remarks on nilpotent orbits.*Journal of Algebra***64**, 190-213, 1980. MR**81i:17005****[KS96]**F. Knop and S. Sahi, Difference equations and symmetric polynomials defined by their zeroes.*International Math. Research Notes***10**, 473-486, 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, Quasi-symmetric functions and factorial Schur functions. (*preprint*).**[OO97]**A. Okounkov and G. Olshanski, Shifted Jack polynomials, binomial formula, and applications.*Math. Res. Letters***4**, 69-78, 1997. CMP**97:08****[Sah94]**S. Sahi, The spectrum of certain invariant differential operators associated to Hermitian symmetric spaces.*Lie Theory and Geometry*, vol. 123,*Progress in Math.*

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Additional Information

**Chal Benson**

Affiliation:
Department of Mathematics and Computer Science, University of Missouri-St. Louis, St. Louis, Missouri 63121

Email:
benson@arch.umsl.edu

**Gail Ratcliff**

Affiliation:
Department of Mathematics and Computer Science, University of Missouri-St. Louis, St. Louis, Missouri 63121

Email:
ratcliff@arch.umsl.edu

DOI:
https://doi.org/10.1090/S1088-4165-98-00040-5

Received by editor(s):
October 22, 1997

Received by editor(s) in revised form:
February 17, 1998

Published electronically:
April 1, 1998

Article copyright:
© Copyright 1998
American Mathematical Society