Hermitian Clifford analysis and its connections with representation theory

Abstract

This work reconsiders the holomorphic and anti-holomorphic Dirac operators of Hermitian Clifford analysis to determine whether or not they are the natural generalization of the orthogonal Dirac operator to spaces with complex structure. We argue the generalized gradient construction of Stein and Weiss based on representation theory of Lie groups is the natural way to construct such a Dirac-type operator because applied to a Riemannian spin manifold, it provides the Atiyah–Singer Dirac operator. This method, however, does not apply to these Hermitian Dirac operators because the representations of the unitary group used are not irreducible, causing problems in considering invariance under a group larger than U(n). This motivates either the development of Clifford analysis over a complex vector space with respect to a Hermitian inner product or the development of Dirac-type operators on Cauchy–Riemann structures.