Highest weight Harish-Chandra supermodules and their geometric realizations

In this paper we 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 $frak g$ (${frak g} eq A(n,n)$), and the maximal compact subalgebra $frak k_r$ of $frak g_{r,0}$, together with their geometric global realizations. These representations occur, as in the ordinary setting, in the superspaces of sections of holomorphic super vector bundles on the associated Hermitian superspaces $G_r/K_r$.