Charles F. Dunkl

Integral kernels with reflection group invariance

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 [4,5,6,7]. 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 measure algebra of a locally compact hypergroup

A hypergroup is a locally compact space on which the space of finite regular Borel measures has a convolution structure preserving the probability measures. This paper deals only with commutative hypergroups.-§1 contains definitions, a discussion of invariant measures, and a characterization of idempotent probability measures. §2 deals with the characters of a hypergroup. §3 is about hypergroups, which have generalized translation operators (in the sense of Levitan), and subhypergroups of such. In this case the set of characters provides much information. Finally §4 discusses examples, such as the space of conjugacy classes of a compact group, certain compact homogeneous spaces, ultraspherical series, and finite hypergroups. A hypergroup is a locally compact space on which the space of finite regular Borel measures has a convolution structure preserving the probability measures. Such a structure can arise in several ways in harmonic analysis. Two major examples are furnished by the space of conjugacy classes of a compact nonabelian group, and by the two-sided cosets of certain nonnormal closed subgroups of a compact group. Another example is given by series of Jacobi polynomials. The class of hypergroups includes the class of locally compact topological semigroups. In this paper we will show that many well-known group theorems extend to the commutative hypergroup case. In §1 we discuss some basic structure and determine the idempotent probability measures. In §2 we present some elementary theory of characters of a hypergroup. In §3 we look at a restricted class of hypergroups, namely those on which there is a generalized translation in the sense of Levitan ([11], or see [12, p. 427]). (The notation of the present paper would seem to have two advantages over Levitans: ours is compatible with current notation for compact groups, and in Levitans notation, it is almost impossible to express correctly commutativity and associativity.) The theory for these hypergroups looks much like locally compact abelian group theory, yet covers a much wider range of examples. Finally in §4 we discuss some examples and further questions. 1. Basic properties. We recall some standard notation (in the following, X is a locally compact Hausdorff space): Received by the editors January 13, 1972. AMS (MOS) subject classifications (1970) Primary 22A20, 22A99, 43A10; Secondary 33A65, 42A60.

Dunkl operators for complex reflection groups

Dunkl operators for complex reflection groups are defined in this paper. These commuting operators give rise to a parameterized family of deformations of the polynomial De Rham complex. This leads to the study of the polynomial ring as a module over the ?rational Cherednik algebra?, and a natural contravariant form on this module. In the case of the imprimitive complex reflection groups

Singular polynomials for finite reflection groups

G(m, p, N)

Orthogonal Polynomials of Types A and B and Related Calogero Models

, the set of singular parameters in the parameterized family of these structures is described explicitly, using the theory of non-symmetric Jack polynomials.

Intertwining operators and polynomials associated with the symmetric group

The Dunkl operators involve a multiplicity function as parameter. For generic values of this function the simultaneous kernel of these operators, acting on polynomials, is equal to the constants. For special values, however, this kernel is larger. We determine these singular values completely and give partial results on the representations of G that occur in this kernel.

An Addition Theorem for Hahn Polynomials: The Spherical Functions

Abstract: There are examples of Calogero–Sutherland models associated to the Weyl groups of type A and B. When exchange terms are added to the Hamiltonians the systems have non-symmetric eigenfunctions, which can be expressed as products of the ground state with members of a family of orthogonal polynomials. These polynomials can be defined and studied by using the differential-difference operators introduced by the author in Trans. Am. Math. Soc. 311, 167–183 (1989). After a description of known results, particularly from the works of Baker and Forrester, and Sahi; there is a study of polynomials which are invariant or alternating for parabolic subgroups of the symmetric group. The detailed analysis depends on using two bases of polynomials, one of which transforms monomially under group actions and the other one is orthogonal. There are formulas for norms and point-evaluations which are simplifications of those of Sahi. For any parabolic subgroup of the symmetric group there is a skew operator on polynomials which leads to evaluation at (1,1,… ,1) of the quotient of the unique skew polynomial in a given irreducible subspace by the minimum alternating polynomial, analogously to a Weyl character formula. The last section concerns orthogonal polynomials for the type B Weyl group with an emphasis on the Hermite-type polynomials. These can be expressed by using the generalized binomial coefficients. A complete basis of eigenfunctions of Yamamotos BN spin Calogero model is obtained by multiplying these polynomials by the ground state.

Weakly almost periodic functionals on the Fourier algebra

AbstractThere is an algebra of commutative differential-difference operators which is very useful in studying analytic structures invariant under permutation of coordinates. This algebra is generated by the Dunkl operators

An addition theorem for someq-Hahn polynomials

T_i : = \frac{\partial }{{\partial x_i }} + k\sum\nolimits_{j \ne i} {\frac{{1 - (ij)}}{{x_i - x_j }}}