Polynomial solutions of the Hermitian submonogenic system

The Hermitian monogenic system is an overdetermined system of two Dirac type operators in several complex variables generalizing both the holomorphic system and the real Dirac system. Due to the fact that it is overdetermined, the Cauchy–Kowalevski extension problem only has a solution if the Cauchy data satisfy certain constraints. There is however a subsystem, called Hermitian submonogenic system, for which theses constraints are no longer necessary, while, if the constraints hold, the Cauchy–Kowalevski extension will still be Hermitian monogenic. In this paper we focus on Cauchy–Kowalevski extensions of general polynomials, in the case of the Hermitian submonogenic system, and we compute the corresponding dimensions and reproducing kernels.