  In this series of papers we want to discuss the highest weight ${\frak
k}_r$-finite representations of the pair $({\frak g}_r,{\frak k}_r)$ consisting
of ${\frak g}_r$, a real form of a complex basic Lie superalgebra of classical
type ${\frak g}$ (${\frak g}\neq A(n,n)$), and the maximal compact subalgebra
${\frak k}_r$ of ${\frak} g_{r,0}$.
  These representations will be concretely realized through spaces of sections
of holomorphic vector bundles on the associated Hermitian superspaces. In this
part we shall discuss only the infinitesimal theory of the pair $({\frak g}_r,
{\frak k}_r)$. We treat the global theory in subsequent papers of the series.
