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 $\frak g$ (${ \frak g}\neq A(n,n)$), 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 $G_r/K_r$.
Categories: math.RT math-ph math.DG math.MP