Harmonics on the Quantum Euclidean Space Related to the Quantum Orthogonal Group

The aim of this paper is to study harmonic polynomials on the quantum Euclidean space E^N_q generated by elements x_i, i=1,2,...,N, on which the quantum group SO_q(N) acts. The harmonic polynomials are defined as solutions of the equation \Delta_q p=0, where p is a polynomial in x_i, i=1,2,...,N, and the q-Laplace operator \Delta_q is determined in terms of the differential operators on E^N_q. The projector H_m: {cal A}_m\to {\cal H}_{m} is constructed, where {\cal A}_{m} and {\cal H}_m are the spaces of homogeneous of degree m polynomials and homogeneous harmonic polynomials, respectively. By using these projectors, a q-analogue of the classical zonal polynomials and associated spherical polynomials with respect to the quantum subgroup SO_q(N-2) are constructed. The associated spherical polynomials constitute an orthogonal basis of {\cal H}_m. These polynomials are represented as products of polynomials depending on q-radii and x_j, x_{j'}, j'=N-j+1. This representation is in fact a q-analogue of the classical separation of variables. The dual pair (U_q(sl_2), U_q(so_n)) is related to the action of SO_q(N) on E^N_q. Decomposition into irreducible constituents of the representation of the algebra U_q(sl_2)\times U_q(so_n) defined by the action of this algebra on the space of all polynomials on E^N_q is given.