  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.
