  In this article, we define the spherical $\pi$-operator over domains in the
$(n-1)$-D unit sphere $S^n$ of $R^n$ and develop new and analogous results on
the operator it self and its mapping properties. We introduce the spherical
Dirac operator $\Gamma_\alpha$ as an $\alpha$- shift of of $\Gamma_omega$,
where $\Gamma_omega$ is the negative of the wedge (or Grassmann) product of
$\omega$ with that of the Dirac operator $D_\omega$. A gegenbauer polynomial is
used as a Cauchy kernel for the spherical Dirac operator $\Gamma_alpha$.
