The Weyl Algebra, Spherical Harmonics, and Hahn Polynomials

TL;DR

This paper explores the structure of the Weyl algebra using duality techniques, introduces a family of orderings and ${r sl}_2$ actions, and connects radial polynomials to Hahn polynomials, advancing understanding of algebraic and harmonic structures.

Contribution

It introduces a one-parameter family of orderings and ${r sl}_2$ actions on the Weyl algebra, linking radial polynomials to Hahn polynomials and generalizing previous results.

Findings

Defined a family of ordering maps for the Weyl algebra.

Constructed ${r sl}_2$ actions corresponding to these orderings.

Connected radial polynomials to continuous Hahn polynomials.

Abstract

In this article we apply the duality technique of R. Howe to study the structure of the Weyl algebra. We introduce a one-parameter family of ``ordering maps'', where by an ordering map we understand a vector space isomorphism of the polynomial algebra on $\NR^{2d}$ with the Weyl algebra generated by creation and annihilation operators $a_1, ..., a_d, a_1^+, ..., a_d^+$. Corresponding to these orderings, we construct a one-parameter family of ${\fr sl}_2$ actions on the Weyl algebra, what enables us to define and study certain subspaces of the Weyl algebra -- the space of Weyl spherical harmonics and the space of ``radial polynomials''. For the latter we generalize results of Luck and Biedenharn, Bender et al., and Koornwinder describing the radial elements in terms of continuous Hahn polynomials of the number operator.