The [formula omitted] Dirac–Dunkl operator and a higher rank Bannai–Ito algebra

Abstract

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.