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Exceptional theta correspondences via Plancherel formulas for rank one symmetric spaces

Jan Frahm, Quentin Labriet

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.

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 or , whenever it exists. The conformal group contains a natural dual pair , where is essentially the automorphism group of the Jordan algebra and is either , or . The groups that arise in this way include the complex exceptional group of type 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 . This yields a one-to-one correspondence between certain representations of and (a finite cover of) . The representations of that occur in this correspondence are in the support of the Plancherel measure for a rank one symmetric space for , and the proof makes use of the corresponding Plancherel formula.
Paper Structure (21 sections, 27 theorems, 112 equations)

This paper contains 21 sections, 27 theorems, 112 equations.

Table of Contents

  1. Statement of the results.
  2. Method of proof.
  3. Relation to other work.
  4. Minimal representations of conformal groups
  5. The conformal group and its Lie algebra
  6. The minimal representation
  7. Dual pairs and orbits
  8. The dual pair
  9. Orbits of $G$ in $\mathcal{O}$
  10. Structure of OH
  11. Theta correspondence vs. Plancherel formula
  12. Representations of $\mathop{\mathrm{PSL}}\nolimits(2,\mathbb{F})$ and $\mathop{\mathrm{PGL}}\nolimits(2,\mathbb{F})$
  13. Intertwining operators
  14. The Plancherel formula for OH
  15. $V$ Euclidean
  16. ...and 6 more sections

Key Result

Theorem A

Let denote the direct integral decomposition of the left regular representation of $G$ on $L^2(\mathcal{O}_G)$ into irreducible unitary representations of $G$, then the restriction of $\Pi_{\textup{min}}$ to $G\times\widetilde{G'}$ decomposes as with $\theta(\pi)$ an irreducible unitary representation of $\widetilde{G'}$, and the map $\pi\mapsto\theta(\pi)$ is one-to-one (possibly ignoring a set

Theorems & Definitions (51)

  • Theorem A: see Theorem \ref{['thm:AbstractTheta']} and Section \ref{['sec:ExplicitTheta']}
  • Theorem B: see Corollaries \ref{['cor:ExplicitThetaEuclidean']}, \ref{['cor:ExplicitThetaNonEuclidean']} and \ref{['cor:ExplicitThetaComplex']}
  • Lemma 1.1: see FK94
  • Lemma 1.2
  • proof
  • Remark 1.3
  • Lemma 1.4
  • proof
  • Lemma 2.1
  • proof
  • ...and 41 more