Projective cases for the restriction of the oscillator representation to dual pairs of type I

TL;DR

This paper investigates how the oscillator representation behaves when restricted to dual pairs of type I, revealing projectivity in stable cases using Howe's duality.

Contribution

It provides a comprehensive analysis of the restriction of the oscillator representation to dual pairs of type I, identifying projective modules in stable and specific additional cases.

Findings

Modules are projective in the stable range and one additional case.

Duality correspondence by Howe is used to analyze restrictions.

Results enhance understanding of oscillator representation restrictions.

Abstract

For all the irreducible dual pairs of type I $(G,G')$, we analyze the restriction of the oscillator representation as a $(\mathfrak{g}', K')$-module, when $G'$ is the smaller group. For all $(G,G')$ in the stable range, as well as one more case, the modules obtained are projective. We use the duality correspondence introduced by Howe to analyze these restrictions.