Clifford analysis offers a higher dimensional function theory
studying the null solutions of the rotation invariant, vector
valued, first order Dirac operator $\partial$. In the more recent
branch Hermitean Clifford analysis, this rotational invariance has
been broken by introducing a complex structure $J$ on Euclidean
space and a corresponding second Dirac operator $\partial_J$, leading
to the system of equations $\partial f = 0 = \partial_J f$ expressing
so-called Hermitean monogenicity. The invariance of this system is
reduced to the unitary group. In this paper we show that this choice
of equations is fully justified. Indeed, constructing the Howe dual
for the action of the unitary group on the space of all spinor
valued polynomials, the generators of the resulting Lie superalgebra
reveal the natural set of equations to be considered in this
context, which exactly coincide with the chosen ones.