The Z2n Dirac-Dunkl operator and a higher rank Bannai-Ito algebra

The kernel of the Z2n Dirac-Dunkl operator is examined. The symmetry algebra An of the associated Dirac-Dunkl equation on Sn−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 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.