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Integral kernels with reflection group invariance. (English) Zbl 0827.33010

Root systems and Coxeter groups are important tools in multivariable analysis. This paper is concerned with differential-difference and integral operators, and orthogonality structures for polynomials associated to Coxeter groups. For each such group, the structures allow as many parameters as the number of conjugacy classes of reflections. The classical orthogonal polynomials of Gegenbauer and Jacobi type appear in this theory as two-dimensional cases. For each Coxeter group and admissible choice of parameters there is a structure analogous to spherical harmonics which relies on the connection between a Laplacian operator and orthogonality on the unit sphere with respect to a group- invariant measure. The theory has been developed in several papers of the author [Math. Z. 197, 33-60 (1988; Zbl 0626.33007); Trans. Am. Math. Soc. 311, No. 1, 167-183 (1989; Zbl 0652.33004); J. Math. Anal. Appl. 143, No. 2, 459-470 (1989; Zbl 0688.33004)]. In this paper, the emphasis is on the study of an intertwining operator which allows the transfer of certain results about ordinary harmonic polynomials to those associated to Coxeter groups. In particular, a formula and a bound are obtained for the Poisson kernel.
The presentation begins with a quick review of the basic definitions and then some integral identities involving Laguerre polynomials and the Gaussian measure. Next there is a study of the analogy between partial derivatives and the differential-difference operators as applied to inner products on spaces of polynomials. The intertwining operator is then defined and shown to be a bounded linear operator with respect to a useful norm on polynomials (absolutely convergent series of homogeneous parts). The author conjectures that the intertwining operator is a positive integral transform in general (in one dimension, it is a form of Weyl’s fractional integral). The paper ends with a reasonably explicit integral for the Poisson kernel for the ball, a kernel which reproduces certain functions from their boundary values, and some examples coming from the group \(Z_2\), including Gegenbauer and disk polynomials.

MSC:

33C80 Connections of hypergeometric functions with groups and algebras, and related topics
20F55 Reflection and Coxeter groups (group-theoretic aspects)
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