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 R n {{\mathbf {R}}^n} . A commutative set of differential-difference operators, each homogeneous of degree − 1 -1 , is the analogue of the set of first-order partial derivatives in the ordinary theory of spherical harmonics. In the case of R 2 {{\mathbf {R}}^2} and dihedral groups there are analogues of the Cauchy-Riemann equations which apply to Gegenbauer and Jacobi polynomial expansions.