Dirac operators and cohomology for Lie superalgebras

TL;DR

This paper explores Dirac cohomology in the context of Lie superalgebras, specifically focusing on unitary representations of the general linear superalgebra and their connection to nilpotent Lie superalgebra cohomology.

Contribution

It extends Dirac cohomology theory to Lie superalgebras, analyzing unitary representations and their relation to nilpotent Lie superalgebra cohomology.

Findings

Established connections between Dirac cohomology and nilpotent Lie superalgebra cohomology.

Analyzed Dirac cohomology for unitary representations of the general linear superalgebra.

Extended prior results from Lie algebras to Lie superalgebras.

Abstract

Vogan raised the idea of Dirac cohomology to study representations of semisimple Lie groups and Lie algebras. He conjectured that the infinitesimal character of Harish-Chandra modules are determined by their Dirac cohomology. Huang and Pand\v{z}i\'{c} proved this conjecture and initiated the research on Dirac cohomology for Lie superalgebras based on Kostant's results. The aim of the present paper is to study Dirac cohomology of unitary representations for the general linear superalgebra and its relation to nilpotent Lie superalgebra cohomology.