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Towards the Relative Langlands Duality for Orthosymplectic Pairs

This paper proves a conjectured categorical equivalence establishing the S-duality between the orthosymplectic pair $\mathrm{SO}_{2n}\times \mathrm{Sp}_{2n}$ acting on a tensor product space and the dual pair $\mathrm{SO}_{2n+1}\times \mathrm{SO}_{2n}$ acting on the cotangent bundle of $\mathrm{SO}_{2n+1}$ , thereby confirming a specific case of the relative Langlands duality and demonstrating that the Langlands functoriality of the Derived Satake isomorphism for $\mathrm{Sp}_{2n},\mathrm{SO}_{2n}$ is realized via the theta correspondence.

Original authors: Dor Mezer

Published 2026-06-03

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Dor Mezer

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 the universe of mathematics as a vast library containing two different sets of encyclopedias. One set, let's call it the "Geometric Library," describes shapes, movements, and symmetries in physical space. The other set, the "Spectral Library," describes these same things using abstract algebra and numbers.

For a long time, mathematicians have suspected that these two libraries are actually describing the exact same reality, just written in different languages. This idea is called the Langlands Duality. It's like having a dictionary that can perfectly translate a poem about a mountain from English to French, preserving every nuance, emotion, and rhythm.

The Main Discovery
In this paper, the author, Dor Mezer, proves a specific, difficult translation between two very complex "chapters" in these libraries.

Chapter A (The Geometric Side): Imagine a giant, multi-dimensional dance floor where two groups of dancers are performing. One group is the "Orthogonal" dancers (moving in a way that preserves angles, like a rigid rotation), and the other is the "Symplectic" dancers (moving in a way that preserves area, like a fluid swirl). They are dancing on a stage made of a special kind of infinite-dimensional space. The paper studies the rules of this dance.
Chapter B (The Spectral Side): On the other side of the library, there is a different set of dancers: one group of "Orthogonal" dancers and another group of "Orthogonal" dancers, but they are dancing on a stage that looks like the "cotangent bundle" of a sphere (a fancy way of saying they are dancing on a sphere and all the possible directions they could move at any point).

The Big Claim
Mezer proves that Chapter A and Chapter B are actually the same story.

He shows that the complex rules governing the dance of the Orthogonal and Symplectic groups (Chapter A) are mathematically identical to the rules governing the dance of two Orthogonal groups (Chapter B). It's as if he discovered that a complex jazz improvisation by a saxophone and a drum kit is actually the exact same song as a classical piano duet, just played with different instruments.

How He Did It (The Analogy)
To prove this, Mezer didn't just look at the dancers; he looked at the "music" they were playing.

The "Basic Beat" (The Generator): He identified a fundamental "beat" or "note" in the geometric dance (called $E_0$ ). This is like the root note of a chord.
The "Hecke Action" (The Conductor): He showed that when you apply specific musical commands (called Hecke actions) to this basic beat, the resulting music matches perfectly on both sides of the library.
The "Translation Dictionary": He built a bridge between the two sides. He proved that if you take the "basic beat" from the geometric side and translate it using his dictionary, it becomes the "structure sheaf" (the most basic, fundamental object) on the spectral side.

The "Theta Correspondence" Connection
One of the most exciting parts of the paper is what this discovery implies about a famous mathematical tool called the Theta Correspondence.

Think of the Theta Correspondence as a magical radio tower. For decades, mathematicians knew this tower could send signals between the Orthogonal and Symplectic groups. But they weren't sure exactly what the signal meant in the grand scheme of the Langlands program.

Mezer's work proves that this radio tower isn't just sending random noise; it is the exact mechanism that translates the "Geometric Language" into the "Spectral Language." It confirms that the Theta Correspondence is the specific "conductor" that makes the two different dances move in perfect sync.

Why This Matters (Within the Paper)

It solves a puzzle: It confirms a specific prediction about how these mathematical groups relate to each other, filling in a missing piece of the "Relative Langlands Duality" puzzle.
It works for a whole family: While this paper focuses on a specific size ( $2n$ ), the author notes that the method works for a whole family of similar problems (called orthosymplectic pairs), suggesting this isn't just a one-off trick but a general principle.
It unifies concepts: It ties together three major concepts: the geometry of infinite spaces, the algebra of symmetries, and the specific "Theta Correspondence" that links them.

In Summary
Dor Mezer has proven that two seemingly different mathematical worlds—one involving a mix of rigid and fluid symmetries, and the other involving a mix of rigid symmetries in a different configuration—are actually identical. He did this by showing that the "music" of the first world translates perfectly into the "music" of the second, and that a famous mathematical tool (the Theta Correspondence) is the exact translator that makes this possible.

Technical Summary: Towards the Relative Langlands Duality for Orthosymplectic Pairs

Problem Statement
This paper addresses a specific case of the Relative Langlands Duality conjecture, which posits an equivalence between a "geometric" category of D-modules on a spherical variety and a "spectral" category of perfect modules on the cotangent bundle of the Langlands dual group. Specifically, the author investigates the pair involving the group $G = SO_{2n} \times Sp_{2n}$ acting on the symplectic vector space $V = \mathbb{C}^{2n}_- \otimes \mathbb{C}^{2n}_+$ .

The central problem is to establish the S-dual of this pair. While previous works ([6], [14], [8]) had resolved similar cases for $(SO_{2n+1}, Sp_{2n})$ and $(GL_n, GL_m)$ , the orthosymplectic case $SO_{2n} \times Sp_{2n}$ acting on the tensor product space remained to be proven within the framework of the non-polarized version of the conjecture. The goal is to demonstrate that the S-dual pair is $SO_{2n+1} \times SO_{2n}$ acting on the cotangent bundle $T^*SO_{2n+1}$ .

Methodology
The proof proceeds through a multi-step strategy combining geometric representation theory, invariant theory, and categorical algebra:

Geometric Setup and Weil Representation: The author defines the "geometric category" as the category of locally compact, renormalized discrete modules over the completed Weyl algebra $W$ of the loop space $(\mathbb{C}^{2n}_- \otimes \mathbb{C}^{2n}_+)_K$ , denoted $W\text{-mod}_{SO_{2n},O \times Sp_{2n},O,lc}$ . This category carries an action of the Hecke categories associated with $SO_{2n}$ and $Sp_{2n}$ . The "spectral category" is identified as $D^{\text{perf}}_{SO_{2n} \times SO_{2n+1}}(T^*SO_{2n+1})$ .
Hecke Action Identification: A crucial preliminary step (Section 4) involves proving that the Hecke actions of $Rep(SO_{2n+1})$ and $Rep(SO_{2n})$ on the geometric category are compatible. Specifically, the action of the standard representation $U \in Rep(SO_{2n+1})$ is shown to factor through the restriction to $SO_{2n}$ and the corresponding Hecke action. This is achieved by explicit computation of the action on the basic object $E_0$ (the module $W/(V \otimes V')_O$ ) and matching the resulting D-modules with those arising from the $SO_{2n}$ side.
Generation and Deequivariantization: The paper establishes that the object $E_0$ generates the entire geometric category under the Hecke action of $Rep(SO_{2n})$ . This allows the problem to be reduced to an algebraic one: constructing an isomorphism between the deequivariantized endomorphism algebra of $E_0$ (denoted $\mathcal{E}$ ) and the algebra of functions on the spectral side, $Sym(\mathfrak{so}_{2n+1}[-2])$ .
Formality and Localization: Using purity considerations (lifting to mixed twistor D-modules), the author proves that the dg-algebra $\mathcal{E}$ is formal (quasi-isomorphic to its cohomology ring $\mathcal{E}^\bullet$ ). The proof then utilizes the Localization Theorem to analyze the structure of $\mathcal{E}^\bullet$ over the categorical quotient $\Sigma_{2n} \times \Sigma_{2n+1}$ .
Invariant Theory and Pseudo-Slices: To prove the isomorphism $\psi: Sym(\mathfrak{so}_{2n+1}[-2]) \to \mathcal{E}$ , the author constructs a "pseudo-slice" for the action of $SO_{2n}$ on $\mathfrak{so}_{2n+1}^*$ . By analyzing the categorical quotient and showing that the map becomes an isomorphism after base change and localization along this pseudo-slice, the global isomorphism is deduced. The case $n=1$ is handled separately via direct computation.

Key Contributions and Results

Main Theorem (Theorem 1.1): The paper proves an equivalence of categories:
$W\text{-mod}_{SO_{2n},O \times Sp_{2n},O,lc} \cong D^{\text{perf}}_{SO_{2n} \times SO_{2n+1}}(T^*SO_{2n+1})$
This equivalence is compatible with the actions of the respective Hecke categories and maps the basic object $E_0$ to the structure sheaf of $T^*SO_{2n+1}$ .
S-Duality Confirmation (Theorem 1.5): As a direct consequence, the paper confirms that the S-dual of the pair $Sp_{2n} \times SO_{2n} \curvearrowright \mathbb{C}^{2n}_- \otimes \mathbb{C}^{2n}_+$ is $SO_{2n+1} \times SO_{2n} \curvearrowright T^*SO_{2n+1}$ . This validates the non-polarized version of the Relative Langlands Duality conjecture for this specific orthosymplectic case.
Theta Correspondence as Functoriality (Theorem 1.7): The paper demonstrates that the theta correspondence between $Sp_{2n}$ and $SO_{2n}$ realizes the local geometric Langlands functoriality corresponding to the natural inclusion $SO_{2n} \hookrightarrow SO_{2n+1}$ . Specifically, the theta lift functor is shown to be isomorphic to the functorial lift defined by the restriction of equivariance and pushforward in the spectral category (up to a specific involution).
Classification of Irreducibles: The irreducible objects in the geometric category are classified by self-dual signatures, corresponding to the dominant coweights of $SO_{2n}$ .

Significance
The paper claims significance as a concrete realization of the Relative Langlands Duality in a setting that is non-polarizable (the variety $T^*SO_{2n+1}$ is hyperspherical but not polarizable in the traditional sense). By proving the equivalence for the orthosymplectic pair, it extends the scope of the conjecture beyond the cases previously established in the literature.

Furthermore, the work provides a rigorous geometric proof that the theta correspondence is not merely a functorial lift but the specific lift dictated by the Derived Satake isomorphism. The methodology, particularly the use of pseudo-slices and deequivariantized endomorphism algebras, is presented as a framework applicable to the general even orthosymplectic case $osp(2m|2n)$, which the author notes will be the subject of future work. The paper does not claim to solve the general odd orthosymplectic case or the full anomaly cancellation for all hyperspherical varieties, but rather establishes a foundational step for the even case.

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