#### Vol. 308, No. 2, 2020

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

### Sabine J. Lang

Vol. 308 (2020), No. 2, 393–406
##### Abstract

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

##### Keywords
dual pairs, oscillator representation, theta correspondence, duality correspondence
Primary: 11F27
Secondary: 22E50