The ${\mathbb{Z}}_2^n$ Dirac-Dunkl operator and a higher rank Bannai-Ito algebra

The kernel of the ${\mathbb{Z}}_2^n$ Dirac-Dunkl operator is examined. The symmetry algebra ${\mathcal{A}}_n$ of the associated Dirac-Dunkl equation on ${\mathbb{S}}^{n-1}$ 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 ${\mathcal{A}}_n$ 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 ${\mathcal{A}}_n$ is proposed.