Highest weight Harish-Chandra supermodules and their geometric realizations. I. The infinitesimal theory

In this series of papers we want to discuss the highest weight ${\frak k}_r$-finiterepresentationsofthepair$({\frak g}_r,{\frak k}_r)$ consisting of ${\mathfrak{g}}_r$, a real form of a complex basic Lie superalgebra of classical type $\mathfrak{g}$ ($\mathfrak{g}\mathrm{\ne }A(n,n)$), and the maximal compact subalgebra ${\mathfrak{k}}_r$ 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.