Kostant homology formulas for oscillator modules of Lie superalgebras

TL;DR

This paper develops formulas for characters and homology groups of oscillator modules in Lie superalgebras using Howe dualities, extending to classical Lie algebras and introducing new dual pairs.

Contribution

It introduces a systematic method for calculating Kostant homology of oscillator modules via Howe dualities, unifying superalgebra and classical Lie algebra cases.

Findings

Formulas for characters and homology groups of oscillator modules

Recovery of Kostant homology formulas for Hermitian symmetric pairs

Identification of two new reductive dual pairs

Abstract

We provide a systematic approach to obtain formulas for characters and Kostant ${\mathfrak u}$-homology groups of the oscillator modules of the finite dimensional general linear and ortho-symplectic superalgebras, via Howe dualities for infinite dimensional Lie algebras. Specializing these Lie superalgebras to Lie algebras, we recover, in a new way, formulas for Kostant homology groups of unitarizable highest weight representations of Hermitian symmetric pairs. In addition, two new reductive dual pairs related to the above-mentioned ${\mathfrak u}$-homology computation are worked out.