Sakellaridis–Venkatesh Conjecture

• The Sakellaridis–Venkatesh Conjecture is a framework defining harmonic analysis on spherical varieties by associating dual group data to the variety rather than the ambient group.
• It predicts that local L2-spectra, distinguished representations, and global period factorizations are governed by relative dual parameters and explicit transfer operators.
• The framework has been verified in low-rank, exceptional, and categorical cases, extending to refined spectral decompositions and Whittaker period identities.

Searching arXiv for recent and foundational papers on the Sakellaridis–Venkatesh conjecture and related relative Langlands developments. The Sakellaridis–Venkatesh conjecture is a conjectural framework for harmonic analysis on spherical varieties $X$, most often homogeneous spaces $X=H\backslash G$, in which the local $L^2$-spectrum of $X$, the structure of distinguished representations, and the factorization of global periods are governed by dual data attached to $X$ rather than to $G$ alone. In this framework one replaces the usual spectral theory of a reductive group by a “relative” spectral theory of a $G$-variety, attaches to $X$ a dual group $\widehat G_X$ and a distinguished morphism into $\widehat G$, and predicts that both local spectra and global period integrals are controlled by parameters factoring through that morphism (Beuzart-Plessis, 22 Sep 2025).

1. Foundational formulation

In the relative Langlands program, the basic geometric object is a $X=H\backslash G$0-variety $X=H\backslash G$1, usually a homogeneous spherical variety $X=H\backslash G$2. A normal $X=H\backslash G$3-variety $X=H\backslash G$4 is spherical if every Borel subgroup $X=H\backslash G$5 has an open orbit on $X=H\backslash G$6; equivalently, $X=H\backslash G$7 has an open $X=H\backslash G$8-orbit (Beuzart-Plessis, 22 Sep 2025). The foundational work of Sakellaridis and Venkatesh proposes that such varieties should admit a duality theory parallel to the ordinary Langlands program, with $X=H\backslash G$9 playing the role of a relative dual group and a distinguished morphism

$L^2$0

playing the role of a relative $L^2$1-morphism (Beuzart-Plessis, 22 Sep 2025).

The early representation-theoretic form of the conjecture, used in low-rank verification work, is a spectral decomposition of the form

$L^2$2

where $L^2$3 is the Plancherel measure of $L^2$4 and $L^2$5 is a multiplicity space (Gan et al., 2011). In this formulation, two consequences are emphasized: the spectral measure class of $L^2$6 should be absolutely continuous with respect to the pushforward of the Plancherel measure on $L^2$7, and its support should lie in Arthur parameters of $L^2$8 that factor through $L^2$9 (Gan et al., 2011).

The general construction of $X$0 uses the weight lattice $X$1, the valuation cone, the little Weyl group $X$2, and the set of spherical roots $X$3. When $X$4 has no spherical roots of type $X$5, these data define a root datum, hence a connected complex reductive dual group $X$6, and the distinguished morphism exists by the Knop–Schalke theorem quoted in the relative Langlands literature (Beuzart-Plessis, 22 Sep 2025). This establishes the structural core of the conjecture: spherical geometry on $X$7 is expected to determine functorial spectral data on the dual side.

2. Local spectral conjectures

Locally, the conjecture is formulated in terms of the unitary spectrum of $X$8 and the Plancherel decomposition of $X$9. The relative characters $X$0 are the local spectral objects, appearing in a decomposition

$X$1

or equivalently

$X$2

where the support of $X$3 is the $X$4-spectrum of $X$5 (Beuzart-Plessis, 22 Sep 2025). The weak local Sakellaridis–Venkatesh conjecture then predicts

$X$6

with $X$7 ranging over tempered $X$8-parameters into $X$9, pushed forward to Arthur parameters of $G$0 through $G$1 (Beuzart-Plessis, 22 Sep 2025). In the stronger formulation, one sums over pure inner forms $G$2, and the resulting packetwise statement is expected to restore the correct nonvanishing behavior across extended Arthur packets (Beuzart-Plessis, 22 Sep 2025).

A crucial geometric refinement is the role of boundary degenerations $G$3. Under the structural hypotheses emphasized in the relative Langlands literature, Bernstein maps

$G$4

assemble the spectrum of $G$5 from relative discrete spectra of the boundary degenerations, and the corresponding dual groups $G$6 are Levi subgroups of $G$7 (Beuzart-Plessis, 22 Sep 2025). This parallels the ordinary Plancherel decomposition for reductive groups, but with spherical boundary degenerations replacing parabolic subgroups.

In the real case, Delorme established an Archimedean analogue of the Sakellaridis–Venkatesh scattering theorem for real reductive spherical spaces, conditional on an analogue of the discrete-series conjecture. The result reconstructs $G$8 from twisted-discrete spectra of the boundary degenerations $G$9 together with scattering operators $G$0, and gives an isometric description of $G$1 as the invariant subspace of

$G$2

under those scattering operators (Delorme, 2023). This is an analogue rather than a repetition of the original $G$3-adic theorem, but it confirms that the SV architecture extends naturally to the real setting.

3. Global periods and the Whittaker refinement

Globally, the relative Langlands program asks for factorizations of automorphic periods

$G$4

into normalized local factors and special values of automorphic $G$5-functions (Beuzart-Plessis, 22 Sep 2025). In the relative framework, the local input is the collection of relative characters occurring in local Plancherel formulas; after suitable normalization, these local terms are expected to multiply to the square of a global period. In the unramified affine case, the normalized local relative characters are expected to be ratios of local $G$6-values, and the global formula is expected to involve special values of automorphic $G$7-functions, although the general conjecture still leaves certain rational constants unspecified (Beuzart-Plessis, 22 Sep 2025).

The Whittaker case is the sharpest explicit refinement of this general picture. For a quasi-split reductive group $G$8, a Borel $G$9, and a nondegenerate character $X$0 of $X$1, Lapid–Mao study the Whittaker–Fourier coefficient

$X$2

and formulate a precise global identity built from local Whittaker pairings (Lapid et al., 2013). The scalar $X$3 is defined by the global formula

$X$4

and the main conjecture is

$X$5

This is stated explicitly as a strengthening of the Sakellaridis–Venkatesh conjectures in the Whittaker case (Lapid et al., 2013).

The sharpening is threefold. First, it identifies the exact leftover global constant predicted abstractly by the SV framework. Second, it fixes the normalizations through $X$6, $X$7, Tamagawa volume, and normalized local Whittaker pairings. Third, it gives a concrete Ichino–Ikeda-style formula for Whittaker coefficients rather than a purely abstract factorization statement (Lapid et al., 2013). For $X$8, Lapid–Mao prove the conjecture unconditionally: $X$9 which matches the fact that $\widehat G_X$0 for $\widehat G_X$1 (Lapid et al., 2013).

4. Verified cases and model examples

A substantial body of work verifies the Sakellaridis–Venkatesh picture in concrete local and global examples, often by theta correspondence, Bessel–Plancherel formulas, or explicit transfer operators.

Variety or model	Predicted source/dual side	Result
$\widehat G_X$2 low-rank classical and exceptional cases	$\widehat G_X$3 from SV tables	Many rank $\widehat G_X$4, and some rank $\widehat G_X$5, cases verified (Gan et al., 2011)
$\widehat G_X$6, $\widehat G_X$7	$\widehat G_X$8	Relative discrete series shown $\widehat G_X$9-distinguished and $\widehat G$0-elliptic in the known family (Smith, 2018)
$\widehat G$1	$\widehat G$2 or $\widehat G$3	Local Plancherel, transfer, relative character identities, global factorization (Gan et al., 2019)
$\widehat G$4	$\widehat G$5	Local Plancherel and relative character identity; global factorization of $\widehat G$6 (Wan, 2019)
$\widehat G$7	$\widehat G$8	Transfer operator and relative character identity proved via exceptional theta correspondence (Le et al., 24 Jul 2025)

The low-rank verification paper of Gan and Gomez makes the conjecture particularly concrete. Under the simplifying assumption of a map $\widehat G$9, it proves formulas of the form

$X=H\backslash G$00

for many classical Stiefel-type and exceptional spherical varieties, with multiplicity spaces identified by Bessel or Whittaker models (Gan et al., 2011). In particular, this paper verifies the conjecture for many rank $X=H\backslash G$01 cases, and some rank $X=H\backslash G$02 and $X=H\backslash G$03 cases, by showing that the actual $X=H\backslash G$04-spectrum is governed by the predicted smaller group $X=H\backslash G$05 (Gan et al., 2011).

For the linear period

$X=H\backslash G$06

Smith proves the Sakellaridis–Venkatesh parameter condition for the relative discrete series family previously constructed by parabolic induction. In this case

$X=H\backslash G$07

the distinguished morphism is the inclusion

$X=H\backslash G$08

trivial on the Arthur $X=H\backslash G$09-factor, and the local conjecture predicts that relative discrete series should have symplectic and elliptic parameters inside $X=H\backslash G$10 (Smith, 2018). Smith proves exactly this for the explicit family

$X=H\backslash G$11

with

$X=H\backslash G$12

showing that the image of $X=H\backslash G$13 is contained in $X=H\backslash G$14 and is not contained in any proper parabolic subgroup there (Smith, 2018).

The rank-one orthogonal case

$X=H\backslash G$15

is treated through theta correspondence with $X=H\backslash G$16 or $X=H\backslash G$17, depending on the parity of $X=H\backslash G$18. In this setting the paper proves a local Plancherel formula

$X=H\backslash G$19

an explicit transfer operator, and the exact local relative character identity

$X=H\backslash G$20

together with a global factorization of the orthogonal period through normalized local distinguished functionals (Gan et al., 2019).

For

$X=H\backslash G$21

Wan proves a tempered Plancherel decomposition

$X=H\backslash G$22

where $X=H\backslash G$23, and then proves a relative character identity comparing $X=H\backslash G$24 with the Whittaker–torus model

$X=H\backslash G$25

under transfer (Wan, 2019). Globally, the $X=H\backslash G$26-period factors through normalized local $X=H\backslash G$27-periods for automorphic representations obtained by theta lifting from $X=H\backslash G$28 (Wan, 2019).

For the exceptional rank-one spherical variety

$X=H\backslash G$29

Yun and collaborators prove the local transfer and relative character identity predicted by the SV philosophy by realizing the transfer operator through the exceptional theta correspondence

$X=H\backslash G$30

The transfer operator is the explicit iterated Fourier transform

$X=H\backslash G$31

and it satisfies the local relative character identity

$X=H\backslash G$32

The paper also states that this operator agrees with the rank-one transfer formula predicted by Sakellaridis (Le et al., 24 Jul 2025).

5. Relative duality, trace formulas, and categorification

Later developments broaden the conjectural picture from spherical varieties to Hamiltonian varieties and categorical local Langlands. In the Ben-Zvi–Sakellaridis–Venkatesh framework, one studies duality between Hamiltonian varieties $X=H\backslash G$33 for $X=H\backslash G$34 and dual Hamiltonian varieties $X=H\backslash G$35 for $X=H\backslash G$36, with periods on the automorphic side expected to match $X=H\backslash G$37-objects on the spectral side (Lu et al., 26 Apr 2025). A concrete test case is

$X=H\backslash G$38

whose BZSV-predicted dual is

$X=H\backslash G$39

The verification in this case is that the $X=H\backslash G$40-period vanishes on cuspidal representations and on Eisenstein series induced from maximal parabolics other than $X=H\backslash G$41, while for Eisenstein series induced from $X=H\backslash G$42 one obtains a precise BZSV-type period formula involving the Rankin–Selberg factor $X=H\backslash G$43 and its normalized local factors (Lu et al., 26 Apr 2025).

Relative trace formula comparisons also enter the program in a systematic way. For certain strongly tempered spherical varieties, Wang and collaborators propose two families of relative trace formula comparisons predicted by BZSV duality and prove the fundamental lemma and smooth transfer in the $X=H\backslash G$44-adic case. In the six principal examples treated, the dual relative group is rank one, and the local transfer is characterized by equality of orbital integrals after explicit scaling factors; the paper also proposes a conjecture on degenerate Whittaker periods generalizing the Lapid–Mao conjecture (Mao et al., 2023). This situates many classical transfer problems inside a common BZSV framework.

On the categorical side, the unramified Ben-Zvi–Sakellaridis–Venkatesh conjecture predicts, for a smooth affine spherical $X=H\backslash G$45-variety $X=H\backslash G$46,

$X=H\backslash G$47

Assuming this unramified equivalence, a 2025 paper proves a tamely ramified Iwahori-level analogue for the Satake-generated subcategory: $X=H\backslash G$48 which is a categorical Iwahori-Satake theorem under the BZSV unramified conjecture (Lin et al., 29 Oct 2025).

A further categorical local formulation places the normalized period conjecture directly inside Fargues–Scholze local Langlands. In this setting the normalized period object $X=H\backslash G$49 on $X=H\backslash G$50 and the normalized $X=H\backslash G$51-sheaf $X=H\backslash G$52 on $X=H\backslash G$53 are conjectured to correspond under the categorical local Langlands equivalence: $X=H\backslash G$54 This is verified for the Iwasawa–Tate and Hecke periods, conditional on the existence of the categorical local Langlands correspondence for $X=H\backslash G$55 with Eisenstein compatibility (Takaya et al., 5 Jan 2026). These results show that the SV philosophy extends beyond numerical multiplicities and period integrals to categorical boundary objects and normalized sheaf-theoretic period functors.

6. Scope, limitations, and sharper refinements

The phrase “the Sakellaridis–Venkatesh conjecture” does not denote a single isolated statement. The literature presents a hierarchy of local spectral, global period, relative duality, and later categorical conjectures. This suggests that the name is best understood as a program whose structural core is the governance of relative harmonic analysis by dual data attached to $X=H\backslash G$56 rather than by $X=H\backslash G$57 alone (Beuzart-Plessis, 22 Sep 2025).

Several limitations are explicit in the literature. The weak local conjecture gives only packet-level containment for $X=H\backslash G$58: it says that the spectrum should lie in Arthur packets attached to parameters factoring through $X=H\backslash G$59, but it does not specify the actual distinguished members of those packets, nor even guarantee nonvanishing without passing to pure inner forms (Beuzart-Plessis, 22 Sep 2025). Likewise, the general global period conjecture is not yet fully precise: unspecified rational constants remain in the general formulation, and sharper statements are known only in special settings such as the Whittaker case (Beuzart-Plessis, 22 Sep 2025).

For this reason, several major refinements should be viewed as sharper relatives rather than replacements. Lapid–Mao identify the precise packet-theoretic constant in the Whittaker case (Lapid et al., 2013). Gan–Gross–Prasad and Prasad-type conjectures determine the actual distinguished member inside a packet in specific families and provide explicit multiplicity formulas in terms of component groups and fibers of the relative $X=H\backslash G$60-group map (Beuzart-Plessis, 22 Sep 2025). In this sense the original SV conjecture often functions as a packet-level or support-level prediction, while later refinements resolve the finer internal structure of the packets.

There are also technical restrictions. The original smooth asymptotic and Bernstein-map theory was developed first for split $X=H\backslash G$61-adic groups and, for some parts, under wavefront hypotheses; extensions to nonsplit groups, arbitrary spherical varieties, and Archimedean settings require additional geometry and analysis (Beuzart-Plessis, 22 Sep 2025). Delorme’s real scattering theorem is conditional on an Archimedean analogue of the SV discrete-series conjecture, although it avoids the wavefront and split assumptions of the original $X=H\backslash G$62-adic theorem (Delorme, 2023). On the categorical side, many statements remain conditional on ambient versions of categorical local Langlands or on unramified BZSV equivalences (Lin et al., 29 Oct 2025).

The enduring content of the conjecture is therefore structural rather than merely enumerative. It predicts that the harmonic analysis of a spherical variety $X=H\backslash G$63, the decomposition of $X=H\backslash G$64, the distinction problem $X=H\backslash G$65, and the factorization of global periods are manifestations of a single relative duality theory. The accumulated evidence—from low-rank Plancherel formulas and theta correspondences to explicit Whittaker constants, exceptional transfer operators, Hamiltonian duality, and categorical local formulations—shows that this relative duality principle is highly rigid across classical, exceptional, local, global, and categorical settings (Gan et al., 2011).