Differential-difference operators associated to reflection groups

Abstract:

There is a theory of spherical harmonics for measures invariant under a finite reflection group. The measures are products of powers of linear functions, whose zero-sets are the mirrors of the reflections in the group, times the rotation-invariant measure on the unit sphere in ${{\mathbf {R}}^n}$. A commutative set of differential-difference operators, each homogeneous of degree $-1$, is the analogue of the set of first-order partial derivatives in the ordinary theory of spherical harmonics. In the case of ${{\mathbf {R}}^2}$ and dihedral groups there are analogues of the Cauchy-Riemann equations which apply to Gegenbauer and Jacobi polynomial expansions.