## Combinatorics and spherical functions on the Heisenberg group

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- by Chal Benson and Gail Ratcliff
- Represent. Theory
**2**(1998), 79-105 - DOI: https://doi.org/10.1090/S1088-4165-98-00040-5
- Published electronically: April 1, 1998
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## 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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## Bibliographic 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
- © Copyright 1998 American Mathematical Society
- Journal: Represent. Theory
**2**(1998), 79-105 - MSC (1991): Primary 22E30, 43A55
- DOI: https://doi.org/10.1090/S1088-4165-98-00040-5
- MathSciNet review: 1613063