Exceptional theta correspondences via Plancherel formulas for rank one symmetric spaces

TL;DR

This paper establishes explicit correspondences between certain representations of minimal groups associated with Jordan algebras and rank one symmetric spaces, using Plancherel formulas to analyze their decompositions.

Contribution

It explicitly determines the direct integral decomposition of minimal representations restricted to dual pairs, revealing new exceptional theta correspondences via Plancherel formulas.

Findings

Explicit decomposition of minimal representations for specific dual pairs

Identification of a one-to-one correspondence between representations of G and G'

Connection of these representations to the Plancherel measure of rank one symmetric spaces

Abstract

We consider the minimal representation of (a finite cover of) the conformal group of a simple split Jordan algebra over $\mathbb{R}$ or $\mathbb{C}$, whenever it exists. The conformal group contains a natural dual pair $G\times G'$, where $G$ is essentially the automorphism group of the Jordan algebra and $G'$ is either $\operatorname{PSL}(2,\mathbb{R})$, $\operatorname{PGL}(2,\mathbb{R})$ or $\operatorname{PGL}(2,\mathbb{C})$. The groups $G$ that arise in this way include the complex exceptional group of type $F_4$ as well as its compact and split real form. We explicitly determine the direct integral decomposition of the minimal representation restricted to the corresponding cover of $G\times G'$. This yields a one-to-one correspondence between certain representations of $G$ and (a finite cover of) $G'$. The representations of $G$ that occur in this correspondence are in the support of the Plancherel measure for a rank one symmetric space for $G$, and the proof makes use of the corresponding Plancherel formula.