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

Abstract

For all the irreducible dual pairs of type I $(G,G^{\prime })$, we analyze the restriction of the oscillator representation as a $({\mathfrak{g}}^{\prime },K^{\prime })$-module, when $G^{\prime }$ is the smaller group. For all $(G,G^{\prime })$ 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.