Exceptional dual pair correspondences; case of real groups of split rank one

Abstract

Exceptional real groups have quaternionic forms of split rank 4 that contain dual pairs $G\times G'$, where $G'$ is the split Lie group of the type $G_2$, and $G$ a Lie group of split rank one. In this paper we restrict the minimal representation of the quaternionic group to the dual pair and prove some significant results for the resulting correspondence of representations.