A family of Spin(8) dual pairs: the case of real groups

TL;DR

This paper explores dual pairs involving Spin(8) within exceptional groups of type E6, analyzing the representation correspondence derived from the minimal representation restriction.

Contribution

It characterizes the dual pairs in E6 containing Spin(8) and describes the associated representation correspondence.

Findings

Identification of dual pairs in E6 involving Spin(8) and a torus with involution

Description of the representation correspondence from the minimal representation

Insight into the structure of these dual pairs within exceptional groups

Abstract

Exceptional groups of type $E_6$ contain dual pairs where one member is $\mathrm{Spin}(8)$, and the other is $T\rtimes \mathbb Z/2\mathbb Z$, where $T$ is a two-dimensional torus and the non-trivial element in $\mathbb Z/2\mathbb Z$ acts on $T$ by the inverse involution. We describe the correspondence of representations arising by restricting the minimal representation.