Differential-difference operators associated to reflection groups

Dunkl, Charles F.

doi:10.1090/S0002-9947-1989-0951883-8

Differential-difference operators associated to reflection groups
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by Charles F. Dunkl

Trans. Amer. Math. Soc. 311 (1989), 167-183

DOI: https://doi.org/10.1090/S0002-9947-1989-0951883-8

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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.

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On quantitative substitutional analysis

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