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

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