🔢 mathematics

Generalised Whittaker models as instances of relative Langlands duality II: Plancherel density and global periods

This paper verifies the numerical conjectures of Ben-Zvi, Sakellaridis, and Venkatesh regarding local Plancherel density and global periods for a general family of relative Langlands duality instances previously proposed by the authors in the context of branching problems.

Original authors: Wee Teck Gan, Bryan Peng Jun Wang

Published 2026-06-04

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Wee Teck Gan, Bryan Peng Jun Wang

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

The Big Picture: A Cosmic Translation Service

Imagine the universe of mathematics has two different languages. One language describes shapes and symmetries (like how a snowflake rotates or how a group of dancers moves together). The other language describes numbers and frequencies (like the specific notes in a song or the values of complex equations).

For a long time, mathematicians knew these two languages were related, but they didn't have a perfect dictionary to translate between them. This paper is about building a very specific, high-precision part of that dictionary.

The authors, Wee Teck Gan and Bryan Wang Peng Jun, are testing a grand theory called the Relative Langlands Duality. Think of this theory as a "Rosetta Stone" that claims:

If you have a specific geometric shape (a "variety") in Language A, there is a matching "dual" shape in Language B.
The "vibrations" (mathematical frequencies) of the first shape should perfectly match the "values" (special numbers) of the second shape.

The Problem: Missing the Fine Print

In a previous paper, the authors showed that this "Rosetta Stone" works for the general structure of these shapes. They proved the types of shapes match up.

However, the previous work was like looking at a map from a high altitude. It showed the continents, but not the specific elevation of every hill. The current paper zooms in to check the exact numbers. Specifically, they wanted to verify two things:

The Density of Sound (Plancherel Density): How "loud" or frequent are the vibrations of a specific shape?
The Global Echo (Global Periods): If you send a signal through the whole system, does the echo match the predicted mathematical value?

The Method: The "Theta Correspondence" Bridge

To solve this, the authors use a tool called Theta Correspondence.

The Analogy: Imagine two rooms, Room A and Room B, separated by a thick wall. You can't see into the other room. However, there is a special, magical bridge connecting them.

If you drop a stone in Room A, it creates a ripple that travels across the bridge and creates a specific ripple in Room B.
The authors use this bridge to take a complex problem in Room A (calculating the density of vibrations) and translate it into a problem in Room B, where the math is easier to solve. Once they solve it in Room B, they translate the answer back to Room A.

In this paper, the "rooms" are groups of symmetries (Orthogonal groups and Symplectic groups), and the "bridge" is a mathematical process called theta lifting.

The Journey of the Paper

1. Setting the Stage (The "Hook" Shapes)
The authors focus on a specific family of shapes that look like "hooks" (mathematically, these are related to specific patterns called partitions). They pick a representative example involving even-dimensional spheres (Orthogonal groups) and show that these shapes have a perfect dual partner.

2. The Spectral Decomposition (Breaking the Sound Down)
They look at a space called $L^2(X)$ . Imagine this space as a giant, complex sound wave. The goal is to break this wave down into its pure, individual notes (frequencies).

The Discovery: They prove that the "notes" in this complex space are exactly the same as the "notes" coming from the dual group, just shifted slightly by the bridge. This confirms that the "sound" of one shape is indeed the "translation" of the sound of its dual.

3. The Basic Function (The Pure Tone)
They focus on a "basic function," which is like a pure, single tone (the simplest possible vibration). They calculate the exact "volume" (density) of this tone.

The Result: When they calculate this volume, it matches the "special number" (L-value) predicted by the dual shape. It's like tuning a guitar string: they calculated the tension, and it matched the note the theory said it should be. This confirms the numerical conjecture for the local density.

4. The Global Echo (The Long-Distance Call)
Finally, they look at the "Global Period." Imagine sending a message from one end of the universe to the other.

They show that if you send a specific type of signal (an automorphic form) through the system, the strength of the echo you receive at the other end is exactly equal to the product of all the local "volumes" they calculated earlier.
This confirms that the local rules (how the sound works in a small room) add up perfectly to create the global rule (how the sound works across the whole universe).

The "Degenerate" Side Quest

In the final section, they tackle a trickier version of the problem called "Degenerate Whittaker periods."

The Analogy: Imagine the previous sounds were clear, high-pitched notes. These "degenerate" periods are like the low, rumbling bass notes or the silence between the notes.
They show that even for these more complex, "messy" cases, the same bridge (Theta Correspondence) works. They prove that the "local relative characters" (a fancy way of saying the specific rules for how these bass notes behave) are naturally provided by the bridge, solving a puzzle that was previously very difficult to define.

The Bottom Line

This paper doesn't invent a new universe; it verifies the blueprint.

The authors took a grand, theoretical map (the Relative Langlands Duality) and proved that for a specific, important family of shapes, the numbers on the map are correct. They showed that:

The "density" of vibrations in one shape perfectly matches the "values" of its dual shape.
The "echoes" across the entire system are consistent with these local calculations.

They did this by using a "bridge" (Theta Correspondence) to translate difficult problems into easier ones, proving that the mathematical universe is more symmetrical and interconnected than we might have guessed.

Technical Summary: Generalised Whittaker Models as Instances of Relative Langlands Duality II

Problem Statement
This paper addresses specific conjectures within the Relative Langlands Duality framework proposed by Ben-Zvi, Sakellaridis, and Venkatesh ([BZSV]). While the authors' previous work ([GW]) established the duality for smooth representations in the context of branching problems for a general family of hyperspherical varieties, it left unaddressed two critical components of the [BZSV] program:

The precise numerical conjecture for the local Plancherel density (specifically for the "basic function" at unramified places).
The conjecture regarding global periods and their relation to special values of automorphic $L$ -functions.

The authors focus on a representative family of examples involving even orthogonal groups $G = O_{2k}$ and their duals, specifically those associated with nilpotent orbits of "hook type." The goal is to verify that the local Plancherel measure associated with one hyperspherical variety corresponds to the local $L$ -value of its dual, and to apply this to global period formulas.

Methodology
The primary tool employed is the theta correspondence between the odd orthogonal group and metaplectic groups (specifically the dual pair $(O_{2k}, Sp_{2a})$ ). The authors build upon the framework established in [GW2] but introduce a significant methodological shift:

Spectral Decomposition via Theta Correspondence: Unlike previous approaches (e.g., [GW2]) which derived spectral decompositions using methods specific to certain cases, this paper demonstrates the spectral decomposition of $L^2(X)$ (where $X$ is a twisted cotangent bundle associated with the hyperspherical variety) directly through the theta correspondence. This approach is shown to be parallel to the treatment of smooth local and global aspects, offering a more generalizable framework.
Local Analysis: The authors analyze the inner product on $L^2(X)$ by relating it to the Whittaker-Plancherel measure on the dual group $G' = Sp_{2a}$ . They utilize the surjectivity of the map from the Weil representation to the space of functions on $X$ (Theorem 3.1) and justify the interchange of integrals to establish a spectral decomposition formula (Theorem 3.6).
Global Analysis: The global period conjecture is verified by combining the local spectral results with the global theta lifting. The authors assume the Lapid-Mao conjecture for metaplectic groups and the multiplicity one property for the relevant theta lifts.

Key Contributions and Results

Verification of Local Plancherel Density (Theorem 5.6 & 5.7):
The paper proves the numerical conjecture of [BZSV, Proposition 9.2.1] for the hyperspherical dual pair $(M_1, M_2)$ .
- The Plancherel density of the basic function $\phi_0$ on $X$ (associated with $M_2$ ) is shown to be the pushforward of a measure on the maximal compact torus of the dual group $G^\vee_X = O_{2a+1}$ .
- This measure is explicitly given by a ratio of determinants involving the representation $V_X$ (associated with $M_1$ ) and the Lie algebra of the dual group.
- The result confirms that the local Plancherel density associated with $M_2$ corresponds to a local $L$ -value attached to $M_1$ , specifically of the form $\prod L(i/2, V_X^{(i)})$ .
Global Period Conjecture (Theorem 6.3):
The authors establish the numerical conjecture for global periods for cuspidal automorphic forms on $G=O_{2k}$ that are in the image of global theta lifting from $G'=Sp_{2a}$ .
- They show that the square of the global period functional $|P_\gamma(f)|^2$ factorizes into a product of local factors, a central $L$ -value term, and a group order factor $|S_\phi|^{-1}$ .
- This result holds under the assumption of the Lapid-Mao conjecture for the Whittaker period on symplectic groups.
Degenerate Whittaker Periods (Section 7):
The paper addresses conjectures by [MWZ] concerning "degenerate Whittaker periods" (where the subgroup $H$ is trivial).
- By utilizing the theta correspondence, the authors define "local relative characters" naturally, resolving a key obstacle noted in [MWZ].
- They prove [MWZ, Conjectures 1.4 and 1.8] for the specific "hook type" cases considered.
- Theorem 7.10 and 7.11 provide explicit formulas for the degenerate Whittaker periods and their product with standard periods, confirming the conjectures for rank 1 spherical varieties like $O_{2k-1} \backslash O_{2k}$ as special cases.

Significance and Claims
The authors claim that this work achieves the "full analog" of the results in [GW2] (which were limited to the rank 1 spherical variety $O_k \backslash O_{k+1}$ ) within the broader context of hyperspherical varieties associated with hook type partitions.

Methodological Generality: The paper emphasizes that the method of using theta correspondence to derive spectral decompositions is not only parallel to smooth and global treatments but is also more general. The authors state that this approach can be applied without significant difficulty to other cases in [GW, Section 7] and potentially to unitary dual pairs.
Resolution of Open Issues: The work fills the gaps left by [GW] regarding the Plancherel density and global periods, providing a concrete verification of the [BZSV] duality in a non-trivial setting involving twisted cotangent bundles.
Natural Definition of Relative Characters: In the context of degenerate Whittaker periods, the paper demonstrates that the theta correspondence furnishes the necessary "local relative characters" naturally, offering a pathway to resolve similar conjectures for other spherical varieties (including exceptional cases via exceptional theta correspondences).

The paper concludes modestly, noting that while the methods are robust for the cases considered, further technical details would be required for exceptional theta correspondences and other non-hook type cases.