Differential-difference operators associated to reflection groups

TL;DR: In this article, a theory of spherical harmonics for measures invariant under a finite reflection group is presented, where 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 Rn.

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 Rn . 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 R2 and dihedral groups there are analogues of the Cauchy-Riemann equations which apply to Gegenbauer and Jacobi polynomial expansions. The analysis of orthogonality structures for polynomials in several variables is a problem of vast dimensions. This paper is part of an ongoing program to establish a workable theory for one particular class. The underlying structure is based on finite Coxeter groups: these are finite groups acting on Euclidean space, generated by reflections in the zero sets of a collection of linear functions (the "roots"); the weight functions for the orthogonality are products of powers of these linear functions restricted to the surface of the unit sphere. In addition, the weight function is required to be invariant under the action of the group. The resulting theory has strong similarities to the theory of spherical harmonics; this was established in previous papers of the author [3, 4, 5]. Most notably, a homogeneous polynomial is orthogonal to all polynomials of lower degree if and only if it is annihilated by a certain second-order differential-difference operator. Ordinary partial differentiation acts as an endomorphism on ordinary harmonic functions; the use of such operators and their adjoints leads to recurrence formulas and orthogonal decompositions for harmonic polynomials. In this paper we construct a commutative set of first-order differentialdifference operators associated to the second-order operator previously mentioned. Received by the editors June 1, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 33A45, 33A65, 20H15; Secondary 20C30, 42C10, 51F15.