Differential-difference operators associated to reflection groups

Author:
Charles F. Dunkl

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

MSC:
Primary 33A45; Secondary 20H15, 33A65, 42C10, 51F15

MathSciNet review:
951883

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

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Additional Information

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

Keywords:
Orthogonal polynomials in several variables,
reflection groups,
spherical harmonics,
Gegenbauer polynomials,
Jacobi polynomials

Article copyright:
© Copyright 1989
American Mathematical Society