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

DOI: 10.2140/pjm.2020.308.393

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

Mathematical Subject Classification 2010

Primary: 11F27

Secondary: 22E50

Milestones

Received: 23 January 2019

Revised: 16 April 2020

Accepted: 29 April 2020

Published: 9 December 2020