Exceptional theta correspondences via Plancherel formulas for rank one symmetric spaces
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908881384374272 |
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| author | Frahm, Jan Labriet, Quentin |
| author_facet | Frahm, Jan Labriet, Quentin |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_11401 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Exceptional theta correspondences via Plancherel formulas for rank one symmetric spaces Frahm, Jan Labriet, Quentin Representation Theory Number Theory Primary 22E46, Secondary 43A85 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. |
| title | Exceptional theta correspondences via Plancherel formulas for rank one symmetric spaces |
| topic | Representation Theory Number Theory Primary 22E46, Secondary 43A85 |
| url | https://arxiv.org/abs/2603.11401 |