Representation Theory

ISSN 1088-4165



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 $V$ be a finite dimensional Hermitian vector space and $K$ be a compact Lie subgroup of $U(V)$ for which the representation of $K$ on $\hbox{${\mathbb C}$}[V]$ is multiplicity free. One obtains a canonical basis $\{p_\alpha\}$ for the space $\hbox{${\mathbb C}$}[\hbox{$V_{\mathbb R}$}]^K$ of $K$-invariant polynomials on $\hbox{$V_{\mathbb R}$}$ and also a basis $\{q_\alpha\}$ via orthogonalization of the $p_\alpha$'s. The polynomial $p_\alpha$ yields the homogeneous component of highest degree in $q_\alpha$. The coefficients that express the $q_\alpha$'s in terms of the $p_\beta$'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 $q_\alpha$ completely determine the generic bounded spherical functions for a Gelfand pair obtained from the action of $K$ on a Heisenberg group $H=V\times \hbox{${\mathbb R}$}$.

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Additional Information

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

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

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