The Dirac-Dunkl operator on the 2-sphere associated to the Z 3 2 reflection group is considered. Its symmetries are found and are shown to generate the Bannai-Ito algebra. Representations of the Bannai-Ito algebra are constructed using ladder operators. Eigenfunctions of the spherical Dirac-Dunkl operator are obtained using a Cauchy-Kovalevskaia extension theorem. These eigenfunctions, which correspond to Dunkl monogenics, are seen to support finite-dimensional irreducible representations of the Bannai-Ito algebra.

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Abstract. The kernel of the Z n 2 Dirac-Dunkl operator is examined. The symmetry algebra An of the associated Dirac-Dunkl equation on S n−1 is determined and is seen to correspond to a higher rank generalization of the Bannai-Ito algebra. A basis for the polynomial null-solutions of the DiracDunkl operator is constructed. The basis elements are joint eigenfunctions of a maximal commutative subalgebra of An and are given explicitly in terms of Jacobi polynomials. The symmetry algebra is shown to act irreducibly on this basis via raising/lowering operators. A scalar realization of An is proposed.

The study of spherical harmonics in superspace, introduced in [J. Phys. A: Math. Theor. 40 (2007) 7193-7212], is further elaborated. A detailed description of spherical harmonics of degree k is given in terms of bosonic and fermionic pieces, which also determines the irreducible pieces under the action of SO(m) × Sp(2n). In the second part of the paper, this decomposition is used to describe all possible integrations over the supersphere. It is then shown that only one possibility yields the orthogonality of spherical harmonics of different degree. This is the so-called Pizzetti-integral of which it was shown in [J. Phys. A: Math. Theor. 40 (2007) [7193][7194][7195][7196][7197][7198][7199][7200][7201][7202][7203][7204][7205][7206][7207][7208][7209][7210][7211][7212] that it leads to the Berezin integral.

In this paper, a new class of Cauchy integral formulae in superspace is obtained, using formal expansions of distributions. This allows to solve five open problems in the study of harmonic and Clifford analysis in superspace.MSC 2000 : 58C50, 30G35, 26B20

A higher rank generalization of the (rank one) Racah algebra is obtained as the symmetry algebra of the Laplace-Dunkl operator associated to the Z n 2 root system. This algebra is also the invariance algebra of the generic superintegrable model on the n-sphere. Bases of Dunkl harmonics are constructed explicitly using a Cauchy-Kovalevskaia theorem. These bases consist of joint eigenfunctions of labelling Abelian subalgebras of the higher rank Racah algebra. A method to obtain expressions for both the connection coefficients between these bases and the action of the symmetries on these bases is presented.A HIGHER RANK RACAH ALGEBRA AND THE Z n 2 LAPLACE-DUNKL OPERATOR 3

Abstract. In this paper, a family of radial deformations of the realization of the Lie superalgebra osp(1|2) in the theory of Dunkl operators is obtained. This leads to a Dirac operator depending on 3 parameters. Several function theoretical aspects of this operator are studied, such as the associated measure, the related Laguerre polynomials and the related Fourier transform. For special values of the parameters, it is possible to construct the kernel of the Fourier transform explicitly, as well as the related intertwining operator.

In this paper an extension of Clifford analysis to superspace is given, inspired by the abstract framework of radial algebra. This framework leads to the introduction of the so-called super-dimension, an important parameter appearing in several formulae. Also the relevant differential operators are introduced and their basic properties are proven. Mathematics Subject Classification (2000). Primary 30G35; Secondary 58C50.