  We describe symmetric diffusion operators where the spectral decomposition is
given through a family of orthogonal polynomials. In dimension one, this
reduces to the case of Hermite, Laguerre and Jacobi polynomials. In higher
dimension, some basic examples arise from compact Lie groups. We give a
complete description of the bounded sets on which such operators may live. We
then provide a classification of those sets when the polynomials are ordered
according to their usual degrees
