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The tower property on the genericity of global theta lifts

This paper establishes that the first occurrence of global theta lifts between dual reductive groups preserves genericity by linking $L$ -function analytic properties to special periods, thereby proving the global Gan-Gross-Prasad conjecture for $\SO_{2n+1} \times \SO_{2}$ under specific conditions.

Original authors: Jaeho Haan, Sanghoon Kwon

Published 2026-05-12

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CC BY 4.0

Original authors: Jaeho Haan, Sanghoon Kwon

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a master architect working with two different types of building blocks: Orthogonal blocks (which we'll call "Square Towers") and Symplectic blocks (which we'll call "Round Towers"). In the world of advanced mathematics, these blocks represent complex groups of numbers and symmetries.

For decades, mathematicians have been trying to understand how to build a bridge between a Square Tower and a Round Tower. This bridge is called a Theta Lift. It's a magical process where you take a specific pattern (an "automorphic representation") from a Square Tower and try to project it onto a Round Tower to see if a matching pattern appears there.

The big question has always been: Does the pattern survive the trip? And if it does, does it keep its most special feature, known as being "Generic" (think of this as having a unique, loud, and distinct "voice" that can be heard clearly)?

The "Tower" Problem

About 40 years ago, a mathematician named Steve Rallis discovered a rule about these bridges, which he called the "Tower Property." He found that if you try to build bridges between towers of different sizes, there is a specific "first floor" where the bridge finally works.

If you try to build the bridge on floors below this first floor, nothing happens (the bridge is empty).
If you build it on the first floor, the bridge appears, and it is a strong, stable, and unique structure (it is "cuspidal" and "generic").
If you build it on floors above the first floor, the bridge still exists, but it becomes weak and loses its unique voice (it becomes "non-generic").

What This Paper Does

The authors, Jaeho Haan and Sanghoon Kwon, wanted to answer a very specific question: Does the "first floor" always preserve that special "voice" (genericity)?

In simpler terms: If you take a loud, distinct pattern from a Square Tower and build a bridge to the very first possible Round Tower, will the pattern on the other side still be loud and distinct?

The Answer: Yes. The authors prove that the "first occurrence" (the first time the bridge works) always preserves this special "voice."

How They Proved It (The Detective Work)

To prove this, the authors didn't just look at the bridges; they looked at the sound waves traveling through them.

Listening for Echoes (Periods): They developed a way to listen for specific "echoes" (mathematical integrals called Bessel periods and Fourier-Jacobi periods) inside the towers. If a pattern has a "voice," it leaves a specific echo.
The L-Function Connection: They discovered a deep link between these echoes and a mathematical object called an L-function. You can think of an L-function as a "health report" or a "frequency analyzer" for the pattern.
- If the L-function has a "pole" (a spike or a singularity) at a specific point, it means the pattern is healthy and has a voice.
- If the L-function is smooth and flat, the pattern is silent.
The Proof Strategy: They showed a chain of logic:
- If the bridge works (the lift is non-zero), then the L-function must have a spike.
- If the L-function has a spike, then the pattern must have a "voice" (it is generic).
- Therefore, the first time the bridge works, the pattern must have a voice.

They also tackled a tricky situation where the towers aren't perfectly symmetrical (quasi-split but not split). They had to invent a new way to measure the sound waves (a modified integral) to make sure their logic held up even when the buildings were slightly crooked.

The Big Payoff: The GGP Conjecture

The paper ends with a practical application of their discovery. There is a famous unsolved puzzle in mathematics called the Gan-Gross-Prasad (GGP) Conjecture. It asks: "When can we find a matching pattern between two specific types of towers?"

Using their new proof that "the first bridge always preserves the voice," the authors were able to solve this puzzle for a specific case: connecting a large Odd-Square Tower ( $SO_{2n+1}$ ) to a tiny, split Square Tower ( $SO_2$ ).

They proved that for this specific pair, a matching pattern exists if and only if the "health report" (the L-function) shows a spike. This confirms a major part of the GGP conjecture for this scenario.

Summary in One Sentence

This paper proves that when you build the very first possible bridge between two complex mathematical towers, the unique "voice" of the original pattern is always preserved, and this discovery helps solve a long-standing puzzle about when these patterns can match up.

Technical Summary: The Tower Property on the Genericity of Global Theta Lifts

Problem Statement
The paper investigates the behavior of global theta lifts between dual reductive pairs of classical groups, specifically focusing on the preservation of genericity (the existence of non-vanishing Whittaker periods) across the "Witt tower" of spaces. While Rallis's tower property (established in the 1980s) resolved the questions of cuspidality and non-vanishing for global theta lifts, the specific behavior of genericity—whether the first occurrence of a non-zero lift preserves the generic nature of the source representation—remained a central open question. The authors aim to determine the precise conditions under which the first occurrence of a global theta lift is generic, extending previous results that were often limited to split groups or "strongly generic" representations.

Methodology
The authors employ a synthesis of local and global techniques rooted in the theory of automorphic forms:

Local Analysis (Vanishing of Theta Lifts):
- The paper computes the twisted Jacquet modules of the local Weil representation with respect to specific unipotent subgroups and characters (Bessel and Fourier-Jacobi types).
- By analyzing the action of the Weil representation on mixed models, the authors prove that local theta lifts vanish in certain ranges (specifically when the dimension of the target space is too large relative to the source). This establishes the "first occurrence" bounds.
Global Period Integrals and L-functions:
- The core strategy involves relating the non-vanishing of global theta lifts to the analytic properties of L-functions (specifically poles at $s=1$ or $s=1/2$ ).
- The authors construct Rankin-Selberg integrals (modified from existing literature to suit quasi-split groups) that unfold to represent partial L-functions.
- They establish identities between these global integrals and special Bessel periods (for orthogonal groups) and special Fourier-Jacobi periods (for symplectic/metaplectic groups).
The Rallis Inner Product Formula:
- To link the non-vanishing of the theta lift to the pole of the L-function, the authors utilize the Rallis inner product formula in the first occurrence range.
- Crucially, they incorporate the theory of local doubling zeta integrals developed by Yamana. This allows them to deduce the non-vanishing of the lift implies genericity without requiring the lift itself to be generic a priori, a condition that previous methods (relying on comparing local L-factors) often necessitated.
Handling Quasi-Split vs. Split Groups:
- The paper addresses the technical subtleties arising from working with quasi-split but non-split orthogonal groups ( $SO_{2n}$ where the discriminant is non-trivial), distinguishing their approach from prior work on split groups.
- They utilize restriction and induction functors to transfer results between orthogonal groups ( $O_{2n}$ ) and special orthogonal groups ( $SO_{2n}$ ), proving that genericity is preserved under these operations.

Key Contributions and Results

Generalized Tower Property for Genericity:
The paper establishes a complete description of the genericity of global theta lifts in the Witt tower for dual pairs $(O_{2n}, Sp_{2m})$ and $(O_{2n+1}, \widetilde{Sp}_{2m})$ .
- Theorem 1.2 & 1.3: For a generic cuspidal automorphic representation $\pi$ (or $\sigma$ ), the first occurrence index $l(\pi)$ is tightly bounded (e.g., $n_0 - 1 \le l(\pi) \le n_0$ for $SO_{2n_0}$ ).
- Preservation of Genericity: The first non-zero theta lift $\Theta(\pi)$ is always generic.
- Non-Genericity of Higher Occurrences: Lifts occurring at indices $i \ge l(\pi) + 2$ are non-generic. Lifts at $l(\pi) + 1$ are generic if and only if the first occurrence index is the lower bound of the possible range.
Equivalence of Conditions:
The authors prove a series of equivalences (Theorems 5.2 and 7.1) linking:
- The existence of a pole in the partial L-function $L_S(s, \pi)$ (or $L_S(s, \sigma \times \chi)$ ).
- The non-vanishing of special Bessel or Fourier-Jacobi periods.
- The non-vanishing and genericity of the global theta lift to the first occurrence space.
- The mere non-vanishing of the global theta lift (which implies genericity in the first occurrence range).
Extension to Quasi-Split and Metaplectic Groups:
- The results extend the work of Ginzburg, Rallis, and Soudry (GRS97) and Furusawa (Fu95) from split groups to quasi-split classical groups.
- The paper provides new results for the dual pair $(\widetilde{Sp}_{2m}, SO_{2n+1})$ , a case not previously addressed in the literature regarding genericity.
- It weakens the "strongly generic" assumption required in earlier works, applying to the broader class of generic representations.
Application to the Gan-Gross-Prasad (GGP) Conjecture:
As a direct application, the authors prove the global tempered GGP conjecture for the pair $(SO_{2n+1}, SO_2)$ under the assumption that $SO_2$ is split and the representation is trivial. They show that the non-vanishing of the Bessel period is equivalent to the non-vanishing of the central L-value $L(1/2, \pi)$ .

Significance and Claims
The paper claims to provide a unified framework for understanding the genericity of theta lifts across orthogonal, symplectic, and metaplectic groups. Its significance lies in:

Completing the Rallis Program: It resolves the genericity aspect of the global theta correspondence, complementing Rallis's earlier work on cuspidality.
Methodological Advancement: By utilizing the Rallis inner product formula combined with Yamana's doubling method, the authors bypass the need for the lift to be generically assumed, allowing for a more robust deduction of genericity from non-vanishing.
Broadening Scope: The results apply to quasi-split groups and metaplectic covers, filling gaps in the literature where previous results were restricted to split cases or specific representation types.
GGP Verification: It offers a new proof of the GGP conjecture for a specific case ( $SO_{2n+1} \times SO_2$ ) by leveraging the established link between theta lifts, periods, and L-functions.

The authors note that while their focus is on classical and metaplectic groups, the methods are expected to extend to unitary groups, though they confine the current discussion to the orthogonal and symplectic settings for clarity. They also acknowledge that the non-tempered GGP conjecture for symplectic groups remains a subject for future work.