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

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 .

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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:
http://dx.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