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The Dunkl--Dirac operator is a deformation of the Dirac operator by means of Dunkl derivatives. We investigate the symmetry algebra generated by the elements supercommuting with the Dunkl--Dirac operator and its dual symbol. This symmetry algebra is realised inside the tensor product of a Clifford algebra and a rational Cherednik algebra associated with a reflection group or root system. For reducible root systems of rank three, we determine all the irreducible finite-dimensional representations and conditions for unitarity. Polynomial solutions of the Dunkl--Dirac equation are given as a realisation of one family of such irreducible unitary representations., Comment: v3 40p. Final version accepted in J. Algebra. See v2 for proof of Thm 4.1

In this paper, a new method is developed to obtain explicit and integral expressions for the kernel of the ( κ , a ) -generalized Fourier transform for κ = 0 . In the case of dihedral groups, this method is also applied to the Dunkl kernel as well as the Dunkl Bessel function. The method uses the introduction of an auxiliary variable in the series expansion of the kernel, which is subsequently Laplace transformed. The kernel in the Laplace domain takes on a much simpler form, by making use of the Poisson kernel. The inverse Laplace transform can then be computed using the generalized Mittag–Leffler function to obtain integral expressions. In case the parameters involved are integers, explicit formulas are obtained using partial fraction decomposition. New bounds for the kernel of the ( κ , a ) -generalized Fourier transform are obtained as well.

We consider the symmetry algebra generated by the total angular momentum operators, appearing as constants of motion of the S 3 Dunkl Dirac equation. The latter is a deformation of the Dirac equation by means of Dunkl operators, in our case associated to the root system A 2 , with corresponding Weyl group S 3 , the symmetric group on three elements. The explicit form of the symmetry algebra in this case is a one-parameter deformation of the classical total angular momentum algebra s o ( 3 ) , incorporating elements of S 3 . This was obtained using recent results on the symmetry algebra for a class of Dirac operators, containing in particular the Dirac–Dunkl operator for arbitrary root system. For this symmetry algebra, we classify all finite-dimensional, irreducible representations and determine the conditions for the representations to be unitarizable. The class of unitary irreducible representations admits a natural realization acting on a representation space of eigenfunctions of the Dirac Hamiltonian. Using a Cauchy–Kowalevski extension theorem we obtain explicit expressions for these eigenfunctions in terms of Jacobi polynomials.

The decomposition of polynomials of one vector variable into irreducible modules for the orthogonal group is a crucial result in harmonic analysis which makes use of the Howe duality theorem and leads to the study of spherical harmonics. The aim of the present paper is to describe a decomposition of polynomials in two vector variables and to obtain projection operators on each of the irreducible components. To do so, a particular transvector algebra will be used as a new dual partner for the orthogonal group leading to a generalisation of the classical Howe duality. The results are subsequently used to obtain explicit projection operators and formulas for integration of polynomials over the associated Stiefel manifold., 34 pages

The kernel of the Z 2 n Dirac–Dunkl operator is examined. The symmetry algebra A n 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 Dirac–Dunkl operator is constructed. The basis elements are joint eigenfunctions of a maximal commutative subalgebra of A n 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 A n is proposed.

We study various aspects of the metaplectic Howe duality realized by the Fischer decomposition for the metaplectic representation space of polynomials on R 2 n valued in the Segal–Shale–Weil representation. As a consequence, we determine symplectic monogenics, i.e. the space of polynomial solutions of the symplectic Dirac operator D s .

A superspace version of the Schr\"odinger equation with a delta potential is studied using Fourier analysis. An explicit expression for the energy of the single bound state is found as a function of the super-dimension M in case M is smaller than or equal to 1. In the case when there is one commuting and 2n anti-commuting variables also the wave function is given explicitly., Comment: 4 pages, accepted for publication in Physics Letters A