Relative Langlands Program

• Relative Langlands Program is a framework for analyzing automorphic periods and local distinction using spherical varieties and Hamiltonian spaces.
• It employs dual groups, trace formulas, and categorical methods to link harmonic analysis with Langlands functoriality and special L-function values.
• Recent advances include precise parameter-space multiplicity conjectures and explicit examples from toric and generalized Whittaker models.

The Relative Langlands Program is a framework for studying automorphic periods, local distinction problems, and their connections with Langlands functoriality and special values of automorphic $L$-functions. Its modern formulations replace the isolated pair $(H,G)$ by a spherical variety $X=H\backslash G$, or more generally by a Hamiltonian $G$-space, and seek a spectral description of harmonic analysis on $X$ in terms of Langlands parameters, dual groups, trace formulas, and geometric or categorical structures. In this sense, the theory is “relative” because it is organized by data attached to a quotient, period, boundary condition, or restriction map rather than by the absolute representation theory of $G$ alone (Beuzart-Plessis, 22 Sep 2025, Ben-Zvi et al., 2024). The ordinary Langlands correspondence remains its background dictionary: automorphic representations, dual groups, Galois parameters, and the function-field geometry of shtukas supply the ambient formalism within which relative questions are posed (Glazunov, 2020).

1. Periods, distinction, and spherical varieties

A basic global datum is a period integral

$$P_H(\varphi):=\int_{[H]} \varphi(h)\,dh,\qquad [H]=H(F)\backslash H(\mathbb A),$$

for a cuspidal automorphic form $\varphi$ on a reductive group $G$. The local analogue is distinction: an irreducible representation $\pi\in \mathrm{Irr}(G(k))$ is $(H,G)$0-distinguished when

$(H,G)$1

The relative program interprets both constructions through the homogeneous space $(H,G)$2, and more generally through the harmonic analysis of spherical varieties, namely $(H,G)$3-varieties with an open Borel orbit (Beuzart-Plessis, 22 Sep 2025).

This perspective organizes a wide range of classical phenomena. The standard $(H,G)$4 integral of Jacquet type, Waldspurger’s toric periods, and base change detected by nonvanishing periods all appear as instances in which a period on $(H,G)$5 reflects functorial transfer or a special $(H,G)$6-value. Locally, $(H,G)$7-distinction is reformulated by the nonvanishing of

$(H,G)$8

where $(H,G)$9 is the Schwartz space of $X=H\backslash G$0; for homogeneous $X=H\backslash G$1, this agrees with $X=H\backslash G$2 (Beuzart-Plessis, 22 Sep 2025). In parallel, relative functoriality is formulated not only for reductive groups but for spherical varieties, with reductive groups appearing as special cases (Sakellaridis, 2018).

Spherical geometry provides the finiteness conditions that make such a program workable. For a normal quasi-affine spherical variety, the coordinate ring is multiplicity-free, and the resulting finiteness of local distinction spaces is the geometric shadow of a relative Plancherel theory (Beuzart-Plessis, 22 Sep 2025). This is the conceptual point at which the Relative Langlands Program separates from the absolute theory: the spectral problem is no longer merely “which representations of $X=H\backslash G$3 occur,” but “which representations occur in the harmonic analysis of $X=H\backslash G$4, and with what relative multiplicities.”

2. Dual groups, boundary degenerations, and Hamiltonian duality

For spherical $X=H\backslash G$5, Sakellaridis–Venkatesh attach a package of invariants: the canonical Cartan $X=H\backslash G$6, a weight lattice, a Weyl group $X=H\backslash G$7, spherical roots, and—when there are no type $X=H\backslash G$8 roots—a dual group $X=H\backslash G$9 defined from the corresponding root datum. A distinguished morphism

$G$0

is part of the structure, and boundary degenerations $G$1, indexed by subsets $G$2, play the role of relative Levi data (Beuzart-Plessis, 22 Sep 2025). On the analytic side, Bernstein maps and scattering operators are expected to build the full $G$3-spectrum of $G$4 from the discrete spectra of these boundary degenerations, exactly as Levi subgroups govern the continuous spectrum for $G$5 (Beuzart-Plessis, 22 Sep 2025).

The corresponding local relative spectral conjecture takes the form

$G$6

where $G$7 consists of tempered parameters into $G$8, and $G$9 is built from the Arthur packet attached through $X$0 (Beuzart-Plessis, 22 Sep 2025). In this formulation, the relative dual group does not merely label a transfer; it is the spectral organizing principle for the harmonic analysis of $X$1.

A newer reformulation replaces spherical varieties by graded Hamiltonian $X$2-spaces. The proposal of relative Langlands duality pairs

$X$3

with $X$4 a graded Hamiltonian $X$5-space and $X$6 a Hamiltonian space for the Langlands dual group $X$7. The automorphic quantization of $X$8 is expected to produce periods or relative trace formulas, while the spectral quantization of $X$9 produces $G$0-functions or $G$1-sheaves; the two sides are presented as symmetric outputs of a duality analogous to electric-magnetic duality of boundary conditions in four-dimensional supersymmetric Yang–Mills theory (Ben-Zvi et al., 2024). In this setting, hyperspherical Hamiltonian spaces admit a structure theorem: they arise by Whittaker induction from data

$G$2

and dual spaces are constructed from the corresponding dual subgroup, commuting $G$3, and symplectic representation (Ben-Zvi et al., 2024).

This Hamiltonian viewpoint places generalized Whittaker models into the relative framework as prototypical branching problems. Their quantizations become $G$4-representations whose spectral support should be controlled by the dual Hamiltonian datum, and for orthogonal and symplectic groups the allowed nilpotent partitions are sharply constrained, with hook-type, Shalika-type, and a small number of exceptional cases singled out (Gan et al., 2023).

3. Local parameter spaces and multiplicity conjectures

One of the clearest local formulations concerns the Galois pair $G$5 for a quadratic extension $G$6. In this setting, Prasad formulates a relative local Langlands conjecture classifying distinguished irreducible admissible representations of $G$7 in terms of Langlands parameters and the geometry of parameter spaces (Prasad, 2015). The key extra object is the opposition group $G$8, obtained by twisting $G$9 by the Chevalley involution; for quasi-split $$P_H(\varphi):=\int_{[H]} \varphi(h)\,dh,\qquad [H]=H(F)\backslash H(\mathbb A),$$0, this is the group from which distinguished representations are expected to arise by base change (Prasad, 2015).

The parameter space is

$$P_H(\varphi):=\int_{[H]} \varphi(h)\,dh,\qquad [H]=H(F)\backslash H(\mathbb A),$$1

and the morphism of $$P_H(\varphi):=\int_{[H]} \varphi(h)\,dh,\qquad [H]=H(F)\backslash H(\mathbb A),$$2-groups

$$P_H(\varphi):=\int_{[H]} \varphi(h)\,dh,\qquad [H]=H(F)\backslash H(\mathbb A),$$3

induces a finite map

$$P_H(\varphi):=\int_{[H]} \varphi(h)\,dh,\qquad [H]=H(F)\backslash H(\mathbb A),$$4

The fundamental conjectural principle is that a representation of $$P_H(\varphi):=\int_{[H]} \varphi(h)\,dh,\qquad [H]=H(F)\backslash H(\mathbb A),$$5 is distinguished precisely when its parameter lifts from $$P_H(\varphi):=\int_{[H]} \varphi(h)\,dh,\qquad [H]=H(F)\backslash H(\mathbb A),$$6, and that the multiplicity of invariant linear forms is governed by the fiber of this finite map, corrected by component-group and inner-form data (Prasad, 2015). In the torus case this is fully verified; for $$P_H(\varphi):=\int_{[H]} \varphi(h)\,dh,\qquad [H]=H(F)\backslash H(\mathbb A),$$7, $$P_H(\varphi):=\int_{[H]} \varphi(h)\,dh,\qquad [H]=H(F)\backslash H(\mathbb A),$$8, and unitary groups the conjecture recovers known patterns of multiplicity and base change (Prasad, 2015).

A distinct but complementary local development is categorical and relative from the outset. For a smooth affine spherical $$P_H(\varphi):=\int_{[H]} \varphi(h)\,dh,\qquad [H]=H(F)\backslash H(\mathbb A),$$9-variety $\varphi$0, assuming the unramified relative local Langlands conjecture of Ben-Zvi–Sakellaridis–Venkatesh, together with placidness and a dimension theory for the loop space $\varphi$1, a tamely ramified local relative statement is proved at Iwahori level. The result identifies the Satake-generated subcategory

$\varphi$2

with a spectral category of perfect complexes on a stack built from the relative Langlands dual of $\varphi$3, namely

$\varphi$4

in a slight variant of Devalapurkar’s tamely ramified conjecture (Lin et al., 29 Oct 2025). Conceptually, the Iwahori refinement is encoded by a Grothendieck–Springer factor, while the relative geometry is encoded by $\varphi$5, the dual of $\varphi$6 (Lin et al., 29 Oct 2025).

Together, these two directions show that “relative local Langlands” has at least two precise incarnations: a parameter-space theory of distinction and multiplicity, and a categorical local equivalence for spherical varieties and their loop spaces.

4. Relative trace formulas, functoriality, and direct images

The relative trace formula is the principal comparison tool of the program. For a spherical variety $\varphi$7, the local Plancherel formula decomposes pairings of Schwartz measures through relative characters $\varphi$8, while globally one forms theta series

$\varphi$9

and distributions on automorphic quotients (Sakellaridis, 2018). Relative functoriality is then expressed as comparison between trace formulas for different spherical varieties, replacing classical endoscopic transfer factors by transfer operators or Hankel transforms (Sakellaridis, 2018).

A central innovation is the use of non-standard Schwartz spaces so that spectral decompositions carry the required $G$0-factors. In this formulation, ordinary functoriality, relative functoriality, and functional equations of $G$1-functions all become instances of trace-formula comparison. The transfer operators that appear in examples are often compositions of equivariant Fourier transforms, sufficiently explicit that they are expected to globalize by Poisson summation (Sakellaridis, 2018). The unifying slogan is that the relevant comparison is no longer orbit-by-orbit scalar matching, but operatorial transfer between spaces of test measures.

Frenkel’s geometrization of trace formulas pushes this further. The Arthur–Selberg trace formula is reinterpreted cohomologically, with automorphic functions replaced by sheaves or $G$2-modules on $G$3, and Hecke operators replaced by Hecke functors. Section 7 of that work formulates an explicitly relative geometric trace formula: for a Whittaker-type sheaf $G$4,

$G$5

which geometrizes a relative trace formula whose spectral side is weighted by $G$6 (Frenkel, 2012). This replaces numerical identities by correspondences between cohomology groups and Hom-spaces, and embeds relative trace formulas into the same categorical architecture as geometric Langlands and functoriality (Frenkel, 2012).

A different relative direction, anticipatory rather than standard in the modern spherical-variety sense, is Parshin’s direct-image conjecture. For a proper morphism of a smooth surface onto a smooth curve,

$G$7

he proposes a direct image

$G$8

designed to parallel the sheaf-theoretic functors $G$9 (Parshin, 2013). The conjecture includes projection, duality, base-change, and composition properties, together with the $\pi\in \mathrm{Irr}(G(k))$0-function identity

$\pi\in \mathrm{Irr}(G(k))$1

This is a relative Langlands principle in the literal sense of pushforward along a morphism of schemes: arithmetic data on $\pi\in \mathrm{Irr}(G(k))$2 should become automorphic data on $\pi\in \mathrm{Irr}(G(k))$3 (Parshin, 2013).

5. Geometric and categorical formulations over function fields and locally

The ordinary Langlands correspondence over global fields provides the indispensable background. For a connected reductive group $\pi\in \mathrm{Irr}(G(k))$4 over a global field $\pi\in \mathrm{Irr}(G(k))$5, automorphic forms on $\pi\in \mathrm{Irr}(G(k))$6 are matched with Galois parameters

$\pi\in \mathrm{Irr}(G(k))$7

and in the function-field setting this correspondence is realized geometrically through shtukas (Glazunov, 2020). Local $\pi\in \mathrm{Irr}(G(k))$8-shtukas are described as function-field analogues of $\pi\in \mathrm{Irr}(G(k))$9-divisible groups, moduli stacks of global $(H,G)$00-shtukas are analogues of Shimura varieties, and their cohomology carries both automorphic and Galois information (Glazunov, 2020). This is not itself a formulation of the Relative Langlands Program, but it supplies the geometric infrastructure—bundles, Hecke correspondences, moduli stacks, Tate modules, and local-to-global deformation theory—on which later relative formulations rely (Glazunov, 2020).

A recent function-field program proposes that the correct automorphic category is not the full $(H,G)$01, but the Hecke-lisse or nilpotent-singular-support subcategory

$(H,G)$02

and that its Frobenius trace recovers the classical space of automorphic functions (Gaitsgory, 29 Sep 2025). On the spectral side one uses a restricted stack of $(H,G)$03-local systems

$(H,G)$04

and the global unramified conjecture takes the form

$(H,G)$05

Locally, one passes to a 2-categorical framework of categorical representations of the loop group $(H,G)$06, with a restricted sub-2-category $(H,G)$07, a local trace conjecture, and a local geometric Langlands equivalence

$(H,G)$08

over the punctured disc (Gaitsgory, 29 Sep 2025).

The ramified version is explicitly relative. For a finite set of points $(H,G)$09, one considers $(H,G)$10, the action of local loop groups $(H,G)$11, and an enhanced automorphic object

$(H,G)$12

defined using shtukas and a quotient $(H,G)$13 (Gaitsgory, 29 Sep 2025). The conjectural spectral counterpart is a relative ind-coherent kernel over a restriction map from the open curve to the local discs,

$(H,G)$14

and the final spectral description of the enhanced automorphic object is obtained by pushing forward the dualizing sheaf along the arithmetic restriction map (Gaitsgory, 29 Sep 2025). This formulation is not the period theory of spherical varieties, but it is unmistakably relative: the global object is encoded as living over its local boundary data via a geometric kernel.

6. Singular, toric, and branching-model examples

One of the notable recent developments is the extension of relative duality beyond the smooth or hyperspherical setting. For certain singular affine cones, numerical weak duality continues to hold between automorphic periods and spectral invariants. Two examples are singled out: the nilpotent cone of $(H,G)$15 matrices, related to Ginzburg’s integral representation for the adjoint $(H,G)$16-function of $(H,G)$17, and the nilpotent cone of $(H,G)$18-tensors, related to Garrett’s triple product integral (Chen et al., 2024). In both cases the spectral side is defined by a nonlinear $(H,G)$19-function built from the graded coordinate ring,

$(H,G)$20

and the main theorem establishes numerical weak duality with discrepancy $(H,G)$21 (Chen et al., 2024). The significance is structural: singular orbit closures can still participate in the automorphic/spectral symmetry predicted by the relative program, even when the spectral object is not a standard automorphic $(H,G)$22-function (Chen et al., 2024).

Affine toric varieties provide the most explicit model of this extension. For Langlands dual split tori $(H,G)$23 and $(H,G)$24, dual cones $(H,G)$25 and $(H,G)$26 define toric dual varieties $(H,G)$27 and $(H,G)$28, and weak numerical duality becomes a combinatorial statement about lattice points, orbit closures, and stabilizers (Chen, 2024). The main theorem gives

$(H,G)$29

with discrepancy $(H,G)$30, where $(H,G)$31 and $(H,G)$32 is the grading weight (Chen, 2024). The refined result is orbit-by-orbit: regularized automorphic contributions of torus orbits are matched with regularized spectral contributions of dual fixed orbits, and the framework extends to disconnected stabilizers and toric Deligne–Mumford stacks (Chen, 2024). In this sense, toric geometry supplies a laboratory in which relative duality is literally duality of cones and faces.

Generalized Whittaker models furnish a different class of explicit examples. They are interpreted as branching problems arising from the quantization of hyperspherical Hamiltonian varieties, and the theta correspondence then relates dual branching problems predicted by relative Langlands duality (Gan et al., 2023). For orthogonal and symplectic groups, the permitted nilpotent partitions are tightly constrained, and an infinite family of hook-type examples is shown locally to satisfy the conjectural duality via the theorem of Gomez–Zhu on generalized Whittaker models under theta lifting (Gan et al., 2023). These examples are important because they connect the abstract Hamiltonian formalism directly to classical representation theory.

The present landscape therefore consists of several compatible but non-identical formulations: the spherical-variety program centered on periods and harmonic analysis, the Hamiltonian duality of relative spaces, local multiplicity conjectures through parameter spaces, categorical local equivalences for loop spaces, and function-field kernel constructions through shtukas and restriction maps. What unifies them is the claim that relative automorphic data—periods, distinction, trace formulas, or sheaf-valued automorphic objects—should admit a spectral description controlled by dual groups, dual spaces, or dual parameter stacks, and that this description is as structural for quotients and boundary conditions as the classical Langlands correspondence is for reductive groups themselves.