Exceptional theta correspondences via Plancherel formulas for 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 $G\times G^{\prime }$, where $G$ is essentially the automorphism group of the Jordan algebra and $G^{\prime }$ is either $\mathrm{PSL}(2,\mathbb{R})$, $\mathrm{PGL}(2,\mathbb{R})$ or $\mathrm{PGL}(2,\mathbb{C})$. 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 $G\times G^{\prime }$. This yields a one-to-one correspondence between certain representations of $G$ and (a finite cover of) $G^{\prime }$. 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.

31 pages. Comments are welcome!