Highest weight Harish-Chandra supermodules and their geometric realizations

Abstract

In this paper we discuss the highest weight ${\mathfrak{k}}_r$-finite representations of the pair $({\mathfrak{g}}_r,{\mathfrak{k}}_r)$ consisting of ${\mathfrak{g}}_r$, a real form of a complex basic Lie superalgebra of classical type $\mathfrak{g}$ ($\mathfrak{g}\ne A(n,n)$), and the maximal compact subalgebra ${\mathfrak{k}}_r$ of ${\mathfrak{g}}_{r,0}$, 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 $G_r/K_r$.

This article contains of part of the material originally posted as arXiv:1503.03828 and arXiv:1511.01420. The rest of the material was posted as arXiv:1801.07181 and will also appear in an enlarged version as subsequent posting