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

Full-text PDF Free Access

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 $\mathbb {C}[V]$ is multiplicity free. One obtains a canonical basis $\{p_\alpha \}$ for the space $\mathbb {C}[V_{\mathbb {R}}]^K$ of $K$-invariant polynomials on $V_{\mathbb {R}}$ 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 $H=V\times \mathbb {R}$.

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Additional Information

**Chal Benson**

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

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

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