Highest weight Harish-Chandra supermodules and their geometric realizations

TL;DR

This paper explores highest weight representations of certain real Lie superalgebras and their geometric realizations as sections of holomorphic super vector bundles on Hermitian superspaces, extending classical theories to superalgebra contexts.

Contribution

It introduces a framework for highest weight $rak k_r$-finite representations of real Lie superalgebras and describes their geometric realizations on Hermitian superspaces.

Findings

Classification of highest weight supermodules.

Construction of geometric realizations on superspaces.

Extension of classical representation theory to superalgebras.

Abstract

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 $G_r/K_r$.