  In this paper, the Dirac operator, acting on super functions with values in
super spinor space, is defined along the lines of the construction of
generalized Cauchy-Riemann operators by Stein and Weiss. The introduction of
the superalgebra of symmetries osp(m|2n) is a new and essential feature in this
approach. This algebra of symmetries is extended to the algebra of conformal
symmetries osp(m + 1, 1|2n). The kernel of the Dirac operator is studied as a
representation of both algebras. The construction also gives an explicit
realization of the Howe dual pair osp(1|2) x osp(m|2n) < osp(m + 4n|2m + 2n).
Finally, the super Dirac operator gives insight into the open problem of
classifying invariant first order differential operators in super parabolic
geometries.
