Exceptional theta correspondences via Plancherel formulas for rank one symmetric spaces

1. Problem Statement and Context

The Classical Context:
The classical theta correspondence establishes a one-to-one correspondence between certain representations of two groups $G$ and $G'$ forming a dual pair inside a symplectic group. This is achieved by restricting the metaplectic representation (of the metaplectic group) to the product $G \times G'$. However, this framework is limited to classical groups.

The Gap:
There is a need to extend theta correspondences to exceptional groups and to non-compact dual pairs where both members are non-compact. While some results exist for compact dual pairs or specific non-compact cases, a unified framework for exceptional groups using minimal representations (replacing the metaplectic representation) was lacking. Most existing literature treats these cases individually rather than via a general structural approach.

The Objective:
The authors aim to construct and explicitly determine exceptional theta correspondences for a family of dual pairs $(G, G')$ inside the conformal group of a simple split Jordan algebra. Specifically, they seek to:

1. Relate the restriction of the minimal representation of the conformal group to the Plancherel formula of a rank-one symmetric space associated with $G$.
2. Provide an explicit, one-to-one correspondence between representations of $G$ and a finite cover of $G'$.

2. Mathematical Framework and Methodology

Algebraic Setup:

• Jordan Algebras: Let $V$ be a simple split Jordan algebra over $\mathbb{F} = \mathbb{R}$ or $\mathbb{C}$ with rank $r \geq 2$.
• Groups:
  • $G$: The automorphism group of $V$ (or a finite cover thereof).
  • $G'$: A group isomorphic to $PSL(2, \mathbb{R})$, $PGL(2, \mathbb{R})$, or $PGL(2, \mathbb{C})$, depending on whether $V$ is Euclidean, real non-Euclidean, or complex.
  • $\mathcal{G}$: The conformal group of $V$, containing the dual pair $G \times G'$.
• Minimal Representation ($\Pi_{min}$): The authors utilize the minimal representation of a finite cover of the identity component of the conformal group. This representation exists for all cases except when the conformal group is $SO(p, q)$ with $p+q$ odd.

Methodology:
The proof strategy relies on the $L^2$-model of the minimal representation, constructed previously by Vergne–Rossi, Dvorsky–Sahi, and Kobayashi–Ørsted.

1. Geometric Decomposition:

   • The representation space of $\Pi_{min}$ is $L^2(\mathcal{O})$, where $\mathcal{O} \subset V$ is the orbit of rank-one elements.
   • The authors identify an open dense subset of $\mathcal{O}$ of the form $\mathbb{F}^\times \times \mathcal{O}_G$, where $\mathcal{O}_G = G \cdot c$ is the orbit of a primitive idempotent $c$ under $G$.
   • $\mathcal{O}_G$ is shown to be a rank-one symmetric space for $G$. It is compact if $V$ is Euclidean and non-compact otherwise.

2. Reduction to Plancherel Formula:

   • The restriction of $\Pi_{min}$ to $G$ corresponds to the left-regular representation on $L^2(\mathcal{O})$.
   • Using the decomposition $\mathcal{O} \cong \mathbb{F}^\times \times \mathcal{O}_G$, the authors relate the decomposition of $\Pi_{min}|_{G \times \tilde{G}'}$ to the Plancherel formula for $L^2(\mathcal{O}_G)$.
   • The measure on $\mathcal{O}$ decomposes into a product of the invariant measure on $\mathcal{O}_G$ and a measure on $\mathbb{F}^\times$.

3. Lie Algebra Analysis (Key Lemma):

   • To track the action of $G'$, the authors analyze the Lie algebra representation of $\mathfrak{g}' \cong \mathfrak{sl}(2, \mathbb{F})$.
   • Lemma 3.5 (Key Lemma): They compute the action of the raising operator $E \in \mathfrak{g}'$ on the tensor product space $C^\infty(\mathcal{O}_G) \otimes C^\infty(\mathbb{F}^\times)$. This computation explicitly links the Casimir operator of $G$ acting on the first factor to the differential operators defining the representations of $G'$ on the second factor.

4. Explicit Classification:

   • By making the Plancherel formula for $L^2(\mathcal{O}_G)$ explicit (using literature results for rank-one symmetric spaces), they identify the specific irreducible representations of $G$ appearing in the spectrum.
   • They match the eigenvalues of the Casimir operators (and the second invariant $D$ for complex cases) to determine the corresponding unitary representations of $G'$.

3. Key Contributions and Results

The paper establishes a general theorem (Theorem A) and provides explicit correspondences (Theorem B) for three distinct cases based on the nature of the Jordan algebra $V$.

Theorem A (General Decomposition)

The restriction of the minimal representation $\Pi_{min}$ to the dual pair $G \times \tilde{G}'$ decomposes as a direct integral:
$$\Pi_{min}|_{G \times \tilde{G}'} \simeq \int_{\widehat{G}}^{\oplus} \pi \boxtimes \theta(\pi) \, d\mu(\pi)$$
where $\mu$ is the Plancherel measure for the symmetric space $\mathcal{O}_G$, and $\theta(\pi)$ is an irreducible unitary representation of $\tilde{G}'$. The map $\pi \mapsto \theta(\pi)$ is a bijection (almost everywhere).

Theorem B (Explicit Correspondences)

Case 1: $V$ is Real and Euclidean

• Symmetric Space: $\mathcal{O}_G$ is compact.
• Spectrum of $G$: Discrete. The representations are finite-dimensional irreducible representations $\pi_{k\beta}$ with highest weight $k\beta$ for $k \in 2\mathbb{Z}_{\geq 0}$.
• Correspondence: $\theta(\pi_{k\beta})$ is the holomorphic discrete series representation of a cover of $PSL(2, \mathbb{R})$ with parameter $k + \frac{rd}{2}$.
  • Here $r$ is the rank and $d$ is the dimension of the off-diagonal Peirce spaces.

Case 2: $V$ is Real and Non-Euclidean

• Symmetric Space: $\mathcal{O}_G$ is non-compact.
• Spectrum of $G$: Contains both a continuous part (degenerate principal series $\pi_{\xi, \lambda}$) and a discrete part (Zuckerman's derived functor modules $A_q(k\beta)$).
• Correspondence:
  • Continuous part: $\theta(\pi_{\xi, \lambda})$ is a unitary principal series representation of $PGL(2, \mathbb{R})$ with specific parameters.
  • Discrete part: $\theta(A_q(k\beta))$ is a discrete series representation of $PGL(2, \mathbb{R})$.
  • Special Note: For the specific case $G = SO(p+1, q+1)$ with $p-q \equiv 2 \pmod 4$, there is a shift in the parameter of the principal series ($\xi \to \xi+1$).

Case 3: $V$ is Complex

• Symmetric Space: $\mathcal{O}_G$ is a complex symmetric space.
• Spectrum of $G$: Purely continuous, consisting of degenerate principal series representations $\pi_{m, \lambda}$.
• Correspondence: $\theta(\pi_{m, \lambda})$ is a unitary principal series representation of $PGL(2, \mathbb{C})$.

4. Significance and Impact

1. Unified Framework for Exceptional Groups: Unlike previous works that treated exceptional theta correspondences case-by-case, this paper provides a uniform proof using the theory of Jordan algebras. This approach naturally encompasses exceptional groups like $F_4$, $E_6$, and $E_7$ alongside classical groups.
2. Archimedean Analogue: The results serve as the archimedean (real/complex) analogue to Savin's work on $p$-adic groups, bridging a gap between local fields and the real/complex settings.
3. Explicitness: The paper does not merely prove existence; it provides explicit formulas for the correspondence, linking the structure constants ($r, d$) of the Jordan algebra directly to the parameters of the resulting representations (e.g., the shift in the discrete series parameter).
4. Methodological Innovation: The use of the Plancherel formula for rank-one symmetric spaces to derive theta correspondences is a novel technique. It transforms a difficult representation-theoretic problem into a geometric analysis problem involving the decomposition of $L^2$ spaces on symmetric spaces.
5. New Results: While the Euclidean case reduces to known classical theta correspondences, the non-Euclidean and complex cases yield new exceptional theta correspondences, particularly for non-compact dual pairs where both groups are non-compact.

In summary, the paper successfully generalizes the theta correspondence to a broad class of exceptional and classical groups by leveraging the geometry of minimal representations and the spectral theory of rank-one symmetric spaces.