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 R 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 sl 2 actions on the Weyl algebra, which enables us to define and study certain subspaces of the Weyl algebrathe space of Weyl spherical harmonics and the space of "radial polynomials". For the latter we generalize results of Louck and Biedenharn, Bender et al., and Koornwinder describing the radial elements in terms of continuous Hahn polynomials of the number operator.

We investigate special classes of polynomials in the quantum mechanical position and momentum operators arising from various operator orderings, in particular from the so-called μ-orderings generalizing well-known operator orderings in quantum mechanics such as the Weyl ordering, the normal ordering, etc. Viewing orderings as maps from the polynomial algebra on the phase space to the Weyl algebra generated by the quantum mechanical position and momentum operators we formulate conditions under which these maps intertwine certain naturally defined actions of the Lie algebra . These conditions arise via certain regularities in coefficients defining the orderings which can nicely be described in terms of some combinatorial objects called here ‘inverted Pascal diagrams’. At the end we establish a connection between radial elements in the Weyl algebra and certain polynomials of the ‘number operator’ expressible in terms of the hypergeometric function. This is related to another representation of , realized in terms of difference operators.