Exceptional Theta Correspondence

• Exceptional Theta Correspondence is a framework connecting classical and exceptional groups through explicit theta lifts and the use of minimal representations.
• It constructs exceptional theta series with detailed Fourier expansions that link modular forms to arithmetic data from structures like the Albert algebra.
• The correspondence realizes functorial lifts of Hecke eigenforms to level-one F4 automorphic representations, reinforcing aspects of the Langlands program.

The exceptional theta correspondence refers to a family of correspondences—generalizing the classical theta correspondence—connecting automorphic and representation-theoretic data of exceptional algebraic groups to that of more classical groups, often illuminated through explicit constructions, explicit formulae, and the use of the minimal representation of a larger group. The framework encompasses both local and global settings, leads to explicit functorial lifts between modular forms and automorphic representations, and, in the automorphic context, can have deep ramifications for the Langlands correspondence, the theory of periods and L-values, and explicit constructions of modular forms. The paradigm is exemplified by the correspondence between the split group $\mathbf{F}_4$ and $\mathbf{PGL}_2$ over $\mathbb{Q}$, as in the construction of level one automorphic representations for $\mathbf{F}_4$ as functorial lifts of classical Hecke eigenforms.

1. Dual Pairs and Minimal Representations in the Exceptional Setting

In the exceptional theta correspondence, dual pairs consist of a classical group and an exceptional group embedded in a larger exceptional algebraic group. For instance, the dual pair $(\mathbf{F}_4,\,\mathbf{PGL}_2)$ is realized inside a group $E$ of type $E_7$. The key technical apparatus is the minimal representation $\Pi_\text{min}$ of $E$, which can be realized as a subrepresentation of an induced (degenerate principal series) representation.

The local and global versions of the correspondence are derived by restricting $\Pi_\text{min}$ to $F \times \mathrm{PGL}_2$, allowing a systematic "lift" of automorphic representations from $\mathrm{PGL}_2$ to $\mathbf{F}_4$. Locally, for real and $p$-adic fields, the correspondence sends the lowest weight holomorphic discrete series (or unramified principal series) of $\mathrm{PGL}_2$ to explicit algebraic representations $V_{n\omega_4}$ of $F_4(\mathbb{R})$ or $F_4(\mathbb{Q}_p)$, as established via the analysis of the minimal representation and earlier work of Gross–Savin and Savin–Karasiewicz. Globally, an integral kernel constructed from $\Pi_\text{min}$ allows one to define the theta lift of a modular form $\varphi$ via

$$\Theta_\phi(\varphi)(g) = \int_{[\mathbf{PGL}_2]} \theta(\phi)(g h)\,\overline{\varphi(h)}\,dh,$$

with $\theta(\phi)$ an automorphic realization of $\Pi_\text{min}$ in $L^2([E])$. The resulting global automorphic representation of $\mathbf{F}_4$ has determined local components $\theta(\pi_v)$, matching those expected from the functorial setup and ensuring compatibility across all places (Shan, 31 Jan 2025).

2. Construction of Exceptional Theta Series and Explicit Fourier Expansions

A central achievement in this framework is the construction of "exceptional theta series"—explicit holomorphic modular forms on $\mathrm{SL}_2(\mathbb{Z})$ with Fourier expansions reflecting the arithmetic and representation theory of $\mathbf{F}_4$. The construction utilizes integral structures on the Albert algebra $\mathbb{J}$ (the 27-dimensional exceptional Jordan algebra defining $F_4$) and polynomials $P$ from the polynomial model $V_{n}(\mathbb{J}_{\mathbb{C}})$ of $V_{n\omega_4}$.

The weighted theta series is defined as

$$\vartheta_{J,P}(z) = \sum_{\substack{T\in J \ T \ge 0,\,\mathrm{rank}(T)=1}} \sigma_3(c_J(T))\,P(T)\,q^{\mathrm{Tr}(T)},$$

where the sum is over rank-one, positive-semidefinite elements $T$ of the chosen integral lattice $J \subset \mathbb{J}$, $c_J(T)$ denotes the content (i.e., maximal positive integer scaling $T$ into $J$), $\sigma_3(n) = \sum_{d|n} d^3$, and $\mathrm{Tr}(T)$ is the trace in $\mathbb{J}$. These theta series, combined with appropriate automorphy factors, produce modular forms on $\mathrm{SL}_2(\mathbb{Z})$ of weight $2n + 12$.

An explicit result is that, for a modular form $f_{\Theta(\alpha)}$ associated to automorphic data $\alpha \in \mathcal{A}_{V_{n\omega_4}}$, the Fourier expansion satisfies

$$f_{\Theta(\alpha)}(z) = \frac{1}{|\Gamma_I|}\!\sum_{T\in J_Z^+}\!\sigma_3\bigl(\mathrm{c}_{J_Z}(T)\bigr)\,\alpha_I(T)\, q^{\mathrm{Tr}(T)} + \frac{1}{|\Gamma_E|}\!\sum_{T\in J_E^+}\!\sigma_3\bigl(\mathrm{c}_{J_E}(T)\bigr)\,\alpha_E(T)\, q^{\mathrm{Tr}(T)},$$

where $J_Z, J_E$ are representatives of the two $F_4(\mathbb{R})$-orbits of Albert lattices and $\Gamma_I$, $\Gamma_E$ their respective stabilizers (Shan, 31 Jan 2025).

For $n > 0$, these exceptional theta series are cusp forms, and the families thus constructed span the full space $S_{2n+12}(\mathrm{SL}_2(\mathbb{Z}))$.

3. Functorial Lifts, Modular Forms, and Automorphic Representations

The exceptional theta correspondence realizes a functorial lifting of automorphic representations. Every Hecke eigenform $f$ of weight $2n+12$ for $\mathrm{SL}_2(\mathbb{Z})$ (i.e., with automorphic representation $\pi$ for $\mathrm{PGL}_2$) admits a global theta lift $\Theta(\pi)$—a level-one automorphic representation of $\mathbf{F}_4(\mathbb{A})$ with archimedean component isomorphic to $V_{n\omega_4}$.

The explicit Fourier expansions of the exceptional theta series demonstrate that the new modular forms on $\mathrm{SL}_2(\mathbb{Z})$ indexed by $F_4$ data coincide with the entire space of cusp forms of the same weight: $$\{\vartheta_{J,P}(z)\}_{J,P} \text{ spans } S_{2n+12}(\mathrm{SL}_2(\mathbb{Z})).$$ This explicit theta lift is analogous to the classical construction of modular forms via theta series for quadratic forms, but in the exceptional group context.

Further, by analyzing period integrals—specifically, evaluating the Spin$_9$–period of the theta lift—one obtains

$\mathcal{P}_{\Spin_9}(\Theta_\phi(\varphi)) = \frac{L(\pi, 5/2) L(\pi, 11/2)}{\zeta(4)\zeta(8)} \cdot I_\infty(\phi_\infty, \phi_\infty),$

with $I_\infty$ an explicit archimedean integral and $L(\pi,s)$ the standard $L$-function. This identity confirms both the nonvanishing of the theta lift (thanks to nonzero $L$-values) and the predictions of the conjectures by Sakellaridis and Venkatesh relating periods to special $L$-values (Shan, 31 Jan 2025).

4. Local–Global Compatibility and Internal Structure

Local–global compatibility of the exceptional theta correspondence is ensured: the global theta lift decomposes as a restricted tensor product of the local theta lifts (with the local theory governed by explicit matching of low weight vectors or unramified principal series). The global construction respects the predicted parameters at every place, validating the functorial nature of the lift.

In detail, for each place $v$, the minimal representation restricts to a sum of local theta correspondences $\theta(\pi_v)$, ensuring that the global automorphic representation $\Theta(\pi) = \otimes'_v \theta(\pi_v)$ possesses the correct archimedean and finite part, matching that predicted by the Langlands philosophy.

The minimal representation's seesaw duality properties are instrumental in comparing periods and confirming that such theta lifts realize automorphic representations with prescribed period and cohomological properties.

5. Broader Impact: Level One Automorphic Forms and the Langlands Program

The explicit realization of a family of level one automorphic representations for $\mathbf{F}_4$—parametrized by classical modular forms—substantiates the extension of Langlands functoriality to the exceptional group context. These representations have archimedean component $V_{n\omega_4}$, and their explicit Fourier expansions can be computed and evaluated due to the algebraic structure of the exceptional Jordan algebra and the underlying representation theory.

The exceptional theta correspondence, providing explicit bridges between classical and exceptional groups, yields modular forms for $\mathrm{SL}_2(\mathbb{Z})$ as theta series indexed by the arithmetic data of $F_4$. By evaluating associated $L$-functions and period integrals, it supports the generalized conjectures on the relation between periods, $L$-values, and automorphic multiplicities for spherical varieties in the context of the Langlands program.

Moreover, these results reinforce the viewpoint that explicit theta correspondences—and, in particular, those involving exceptional algebraic groups—are not only accessible to explicit construction, but are also central to the construction and classification of automorphic representations, the realization of functorial lifts (including Arthur packets and A-parameters in more advanced settings), and the arithmetic paper of special values and periods.

6. Technical Tools and Key Formulas

The exceptional theta correspondence as constructed leverages:

• Minimal representation theory (both local and global) of exceptional algebraic groups.
• Explicit integral kernel constructions for theta lifting, combining automorphic kernel functions and dual pair symmetries.
• The use of Albert algebras, their integral lattices, and content/traces, to describe and index representatives for the space of modular forms.
• Explicit formulas for weighted theta series:

$$\vartheta_{J,P}(z) = \sum_{\substack{T \in J \\mathrm{rank}(T)=1}} \sigma_3\bigl(\mathrm{c}_J(T)\bigr) P(T) q^{\mathrm{Tr}(T)},$$

with $q = e^{2\pi i z}$.

• Period relations involving $L$-values:

$\mathcal{P}_{\Spin_9}(\Theta_\phi(\varphi)) = \frac{L(\pi, 5/2) L(\pi, 11/2)}{\zeta(4)\zeta(8)} \cdot I_\infty(\phi_\infty, \phi_\infty).$

These underpin both the local analysis (structure of representations, intertwining operators, parameter matching) and the global analysis (periods, nonvanishing, and $L$-functions).

---

In sum, the exceptional theta correspondence between $\mathbf{F}_4$ and $\mathbf{PGL}_2$ demonstrates the feasibility of explicit functorial lifts from modular forms to automorphic representations of exceptional groups, yields modular forms as exceptional theta series with explicit expansions, and forges new bridges between the arithmetic of modular forms and the automorphic representation theory of exceptional Lie groups (Shan, 31 Jan 2025).