Abstract.For any finite reflection group G acting on RN there is a family of G-invariant measures (h2 du, where A is a certain product of linear functions whose zero-sets are the reflecting hyperplanes for G) on the unit sphere and an associated partial differential operator ( Lhf ■ = A( fh) -/Aft; A is the Laplacian). Analogously to spherical harmonics, there is an orthogonal (with respect to h2 du) decomposition of homogeneous polynomials, that is, if p is of degree n then l«/2]