Reproducing kernels for polynomial null-solutions of Dirac operators

TL;DR

This paper derives the reproducing kernel for the space of polynomial bihomogeneous null-solutions of complex Dirac operators, expressing it via Jacobi polynomials and using Lie superalgebra techniques.

Contribution

It introduces a new explicit form of the reproducing kernel for Hermitian monogenics using Lie superalgebra methods and connects it to classical kernels via algebraic structures.

Findings

Reproducing kernel expressed as sums of Jacobi polynomials

New proof of kernel in the standard Dirac case

Connection between Lie superalgebras and kernel representations

Abstract

It is well-known that the reproducing kernel of the space of spherical harmonics of fixed homogeneity is given by a Gegenbauer polynomial. By going over to complex variables and restricting to suitable bihomogeneous subspaces, one obtains a reproducing kernel expressed as a Jacobi polynomial, which leads to Koornwinder's celebrated result on the addition formula. In the present paper, the space of Hermitian monogenics, which is the space of polynomial bihomogeneous null-solutions of a set of two complex conjugated Dirac operators, is considered. The reproducing kernel for this space is obtained and expressed in terms of sums of Jacobi polynomials. This is achieved through use of the underlying Lie superalgebra $\mathfrak{sl}(1|2)$, combined with the equivalence between the $L^2$ inner product on the unit sphere and the Fischer inner product. The latter also leads to a new proof in the standard Dirac case related to the Lie superalgebra $\mathfrak{osp}(1|2)$.