Differential-difference operators associated to reflection groups. (English) Zbl 0652.33004

Summary: 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 \({\mathbb{R}}^ n\). 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 \({\mathbb{R}}^ 2\) and dihedral groups there are analogues of the Cauchy-Riemann equations which apply to Gegenbauer and Jacobi polynomial expansions.


33C55 Spherical harmonics
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
20F55 Reflection and Coxeter groups (group-theoretic aspects)
42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
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