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

TL;DR

This paper develops the infinitesimal theory of highest weight Harish-Chandra supermodules for certain real Lie superalgebras, laying groundwork for geometric realizations via holomorphic vector bundles on Hermitian superspaces.

Contribution

It introduces a detailed infinitesimal framework for highest weight supermodules associated with real forms of classical Lie superalgebras, extending classical representation theory to the super setting.

Findings

Characterization of highest weight ${rak k}_r$-finite representations

Framework for geometric realization on Hermitian superspaces

Foundations for global theory in subsequent work

Abstract

In this series of papers we want to 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 ${\frak} g_{r,0}$. 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.