Relative Langlands Duality Framework

Updated 1 February 2026

Relative Langlands duality is a framework pairing Hamiltonian G-spaces with dual Ǧ-spaces, connecting automorphic periods and spectral invariants.
It employs methods like Whittaker and symplectic induction along with categorical Satake equivalences to match moment maps and cohomological invariants.
Recent advances have confirmed its arithmetic, geometric, and physical analogies, with examples from Rankin–Selberg duality and S-duality in gauge theory.

Relative Langlands duality generalizes the classical Langlands correspondence by positing a duality relation not just between reductive groups and their representations, but between Hamiltonian G-spaces and their duals, typically hyperspherical or spherical varieties for G and its Langlands dual group Ǧ. This duality governs deep connections between automorphic periods on one side and special values of L-functions or spectral invariants on the other, often recasting multiplicity problems for representations as duality statements between period integrals and spectral data. Recent advances have clarified its arithmetic, geometric, and categorical aspects, and revealed analogies with S-duality in 4d gauge theory.

1. Foundations and Formalism of Relative Langlands Duality

Relative Langlands duality attaches to a Hamiltonian G-space M (often with a polarization, e.g., M = T*X for some G-variety X) a dual Hamiltonian Ǧ-space M̌. The construction begins by encoding the Hamiltonian space M in terms of group-theoretic data: a subgroup H ⊂ G, a commuting Arthur sl₂-action, and auxiliary symplectic H-representations. The “relative” aspect arises since instead of merely a group, one considers a pair (G, X=H\G), or more generally, structures attached to spherical/hyperspherical varieties.

This duality is formally expressed as:

On the automorphic side: analytic periods P_X(φ) = ∫_{[H]\X} φ(x) dx for a cusp form φ on G(𝔸).
On the spectral side: L-functions L_{X̌}(π̌) associated with Galois parameters for (Ǧ, X̌), where X̌ is a dual spherical variety for Ǧ.

The conjectural core is the equivalence

$P_X(φ) \sim L_{X^\vee}(π)$

up to explicit arithmetic normalizations, reflecting an "electric-magnetic" duality in the sense of boundary conditions for supersymmetric gauge theories (Ben-Zvi et al., 2024).

2. Hyperspherical Hamiltonian Varieties and Dual Constructions

A hyperspherical Hamiltonian G-variety M admits a G-equivariant symplectic structure and satisfies:

Poisson-commutative invariant algebra O(M)^G,
Multiplicity-free actions, and
Neutral grading by Gₘ.

From such M, one constructs its dual M̌ for Ǧ via "Whittaker induction" and symplectic induction, using dual data (Ǧ_X⊂Ǧ, Arthur sl₂ parameter, dual symplectic representation S_X) (Ben-Zvi et al., 2024). The duality matches moment maps, centralizer structures, and cohomological invariants, cemented by categorical Satake equivalences, and is compatible with symplectic reduction and convolution structures (Nakajima, 2024).

3. Arithmetic and Geometric Manifestations: Periods and L-values

The duality provides a dictionary:

Automorphic periods and spherical varieties: Integrals of automorphic forms over cycles/subgroups are paired with the geometry of hyperspherical/spherical varieties.
Spectral data and L-functions: L-factors attached to dual groups and representations, often built from explicit dual varieties and their symplectic/Poisson structures.

This is explicitly visible in the case X = Sp₍2n₎\GL₍2n+1₎, whose dual is conjecturally (and now numerically/geometrically verified) to be X̌ = (GLₙ×GLₙ₊₁)\GL₍2n+1₎ (Lu et al., 26 Apr 2025). The periods for Eisenstein series on GL₍2n+1₎ with respect to Sp₍2n₎ match central L-values for representations on GLₙ×GLₙ₊₁, in exact agreement with predictions [BZSV].

Regular quotient theory (Ngô–Morrissey) gives tools for describing centralizer group schemes, structure of moduli spaces, and the flatness/étaleness of Hitchin-type fibrations that underlie geometric incarnations of the duality (Hameister et al., 2024).

4. Categorical and Derived Satake Realizations

The geometric/categorical refinement uses the geometric Satake correspondence and derived Satake categories:

Local categorification: Sheaf categories D^{∞_ℓ(Bun_G)} and mapping stacks QCoh(Loc^{Ǧ}), with period sheaves P_X and L-sheaves L_{X̌} constructed out of functorial mappings and dualizing complexes (Takaya et al., 5 Jan 2026).
Normalized period and L-value conjecture: Under the categorical Langlands equivalence, normalized period sheaves correspond to normalized L-sheaves, and distinction problems become questions about support for these objects (Takaya et al., 5 Jan 2026).
Multiplicative Hitchin fibrations and 1-motives: Duality between twisted/untwisted Hitchin systems, with fiberwise Fourier–Mukai transform matches generic fibers as dual abelian 1-motives (families of commutative group schemes, e.g., Pryms, Picards) (Gallego, 17 Sep 2025).

For symmetric spaces and real groups, relative Satake categories provide equivalences between constructible sheaves on loop symmetric spaces and perverse sheaves on twisted flag varieties, revealing the dual group structures and root-data arising in real forms and symmetric varieties (Chen et al., 2024, Çiloğlu, 2024, Chen, 25 Jan 2026).

5. Examples and Explicit Constructions

Classical Cases

Godement–Jacquet and Whittaker periods: For X=Matₙ, X̌ its Whittaker dual; the relative duality recovers the standard L-function, with categorical sheaf-theoretic matches (Ben-Zvi et al., 2024).
Rankin–Selberg and Gross–Prasad: Friedberg–Jacquet, Jacquet–Ichino, and other branching problems recast as duality between period integrals and special L-values (Hameister et al., 2024, Gan et al., 2024).

Finite Fields and Gelfand Pairs

Finite field instances: Lusztig's Jordan decomposition and canonical correspondences compatible with parabolic induction/theta-lifting realize electric-magnetic duality for periods and multiplicities in finite classical groups (Wang, 2024, Movshev, 2017).
Toric periods and singular spaces: Duality for affine toric varieties, including orbifold and stacky settings, is achieved via explicit character and Frobenius-trace computations (Chen, 2024, Chen et al., 2024).

Non-split and Twisted Settings

Recent work establishes duality between cotangent bundles of quasi-split symmetric spaces and twisted loop symmetric spaces for the dual group, extending derived Satake constructions and Bezrukavnikov equivalences (Chen, 25 Jan 2026, Çiloğlu, 2024).

6. Physical Analogies: Boundary S-duality and Gauge Theory

A distinguishing perspective relates relative Langlands duality to boundary conditions in 4d $\mathcal N=4$ SYM:

S-duality exchanges G and Ǧ, with boundary conditions corresponding to Hamiltonian G-spaces and their duals.
Kapustin–Witten, Gaiotto–Witten, and Coulomb branch algebras provide parallel mathematical structures to the duality conjectures in arithmetic and geometry (Nakajima, 2024, Ben-Zvi et al., 2024).

Mirror symmetry and gauge-theoretic duality inform both the categorical and geometric mechanisms that underlie relative Langlands duality.

7. Open Problems and Future Directions

Work remains to:

Extend duality to ramified, archimedean, and singular varieties,
Systematically classify hyperspherical varieties supporting duality,
Descent from categorical statements to explicit period and Plancherel measure correspondences,
Integrate global trace formulas, categorical distinction, and TFT perspectives,
Generalize to supergroups, quantum Langlands, and broader geometric settings (Ben-Zvi et al., 2024).

Ongoing research continues to clarify combinatorial invariants, geometric/homological structures, and representation-theoretic ramifications entailed by relative Langlands duality.

Table: Representative Constructions and Key Papers

Construction/Concept	Main Reference(s)	Mathematical Setting
Dual Hamiltonian varieties, S-duality	(Nakajima, 2024, Ben-Zvi et al., 2024)	Hamiltonian G-spaces, Boundary gauge theory
Friedberg–Jacquet, Rankin–Selberg duality	(Hameister et al., 2024, Lu et al., 26 Apr 2025)	Spherical/hyperspherical varieties, Hitchin fibration
Relative Satake, real/symmetric varieties	(Chen et al., 2024, Chen, 25 Jan 2026)	Loop spaces, perverse sheaves, derived categories
Categorical period-L correspondence	(Takaya et al., 5 Jan 2026)	Langlands correspondence for stacks
Toric periods, singular duality	(Chen, 2024, Chen et al., 2024)	Toric, stacky, and singular varieties
Finite Gelfand pairs, finite field instances	(Movshev, 2017, Wang, 2024)	Finite groups, representation/character algebra

Relative Langlands duality thus constitutes a unified framework simultaneously linking group theory, automorphic forms, algebraic geometry, representation theory, and physical dualities, and is validated in numerous classical and modern contexts.