Orthogonal Gross–Prasad Periods

• Orthogonal Gross–Prasad periods are period integrals attached to quadratic space inclusions that connect representation restrictions, Bessel models, and central L-values.
• They leverage Bessel and relative Whittaker models to articulate higher codimension integrals, employing explicit unipotent characters and local trace formulas.
• These periods integrate global automorphic representations with local Langlands parameters, underpinned by multiplicity one results and refined Ichino–Ikeda type formulas.

Orthogonal Gross–Prasad periods are period integrals attached to an inclusion of quadratic spaces $(V',q')\subset (V,q)$ and the induced embedding of special orthogonal groups $SO(V',q')\subset SO(V,q)$. In the codimension-one case they are integrals over $SO(V')$; in higher codimension they are realized through Bessel or relative Whittaker models involving a unipotent subgroup and a generic character. The subject links three problems: restriction of representations from a larger orthogonal group to a smaller one, factorization of global periods into local models, and the relation of those periods to central values of Rankin–Selberg type $L$-functions. The local multiplicity theorem for generic $L$-packets of special orthogonal groups was proved by Mœglin and Waldspurger (Moeglin et al., 2010), the tempered local conjecture over all local fields of characteristic $0$ by Luo via a local trace formula (Luo, 2020), and refined global formulas have been established in several orthogonal Bessel settings (Furusawa et al., 2016, Furusawa et al., 2022).

1. Global period integrals and Bessel realizations

The basic global orthogonal Gross–Prasad period for quadratic spaces $(V,q)\supset (V',q')$ over a number field $k$ is the integral

$$\mathcal{P}(\phi,\phi')=\int_{SO(V',k)\backslash SO(V',\mathbb{A})}\phi(g')\,\overline{\phi'(g')}\,dg',$$

for $\phi\in\pi$ and $SO(V',q')\subset SO(V,q)$0, where $SO(V',q')\subset SO(V,q)$1 and $SO(V',q')\subset SO(V,q)$2 are cuspidal automorphic representations of $SO(V',q')\subset SO(V,q)$3 and $SO(V',q')\subset SO(V,q)$4. In the Gross–Prasad–Ichino–Ikeda formalism, non-vanishing of this period is predicted to be equivalent to non-vanishing of a central value $SO(V',q')\subset SO(V,q)$5, subject to local compatibility conditions at every place (Moeglin et al., 2010).

A common misconception is that orthogonal Gross–Prasad periods are only codimension-one restriction integrals. In fact, when $SO(V',q')\subset SO(V,q)$6, the relevant global periods are frequently Bessel periods. In the $SO(V',q')\subset SO(V,q)$7 setting, Furusawa and Morimoto realize the orthogonal period on an inner form $SO(V',q')\subset SO(V,q)$8 as a Bessel period over the subgroup $SO(V',q')\subset SO(V,q)$9,

$SO(V')$0

where $SO(V')$1 and $SO(V')$2 (Furusawa et al., 2022). In the higher-rank special Bessel setting for $SO(V')$3, Furusawa and Morimoto define

$SO(V')$4

and call it special when the representation of $SO(V')$5 is trivial (Furusawa et al., 2016).

These Bessel realizations are not secondary reformulations. They are the natural higher-codimension Gross–Prasad models on orthogonal groups. In particular, they make explicit the unipotent character data that is invisible in the codimension-one integral and provide the local models that appear in refined period formulas.

2. Local multiplicities, Bessel models, and the local Gan–Gross–Prasad problem

Locally, let $SO(V')$6 be a non-archimedean local field of characteristic $SO(V')$7, let $SO(V')$8 and $SO(V')$9 have dimensions $L$0 of opposite parity, and write $L$1, $L$2. When $L$3, the Gross–Prasad multiplicity is

$L$4

For $L$5, Mœglin and Waldspurger replace the ordinary restriction space by a relative Whittaker model. After choosing a unipotent subgroup $L$6 and a character $L$7 invariant under conjugation by $L$8, they define

$L$9

and $L$0 is the dimension of this space; by results of Aizenbud–Gourevitch–Rallis–Schiffmann and Gan–Gross–Prasad one always has $L$1 (Moeglin et al., 2010).

The local conjecture organizes these multiplicities by local Langlands parameters $L$2, $L$3 and their $L$4-packets $L$5, $L$6. The prediction is that there is at most one pair in $L$7 with multiplicity $L$8, and that this pair is selected by explicit epsilon-type invariants. In Mœglin–Waldspurger’s formulation, the decisive datum is the sign $L$9 relative to the quasi-splitness sign

$0$0

If $0$1, all multiplicities vanish; if $0$2, there is a unique distinguished pair in the two packets with multiplicity $0$3 (Moeglin et al., 2010).

Luo gives a complementary tempered formulation for all local fields of characteristic $0$4. For an admissible pair $0$5, with $0$6 and $0$7 in the Bessel case, the local multiplicity is

$0$8

where $0$9 is a generic character of $(V,q)\supset (V',q')$0. The local Gan–Gross–Prasad conjecture asserts that for every generic tempered parameter $(V,q)\supset (V',q')$1,

$(V,q)\supset (V',q')$2

so exactly one representation in the union of tempered packets over all pure inner forms has nonzero Bessel model (Luo, 2020). This local multiplicity is the local indicator for non-vanishing of the local period factor in the global period formula.

3. Generic packets, induction, and the distinguished local pair

For odd orthogonal groups $(V,q)\supset (V',q')$3, an $(V,q)\supset (V',q')$4-parameter is a homomorphism

$(V,q)\supset (V',q')$5

while for even orthogonal groups $(V,q)\supset (V',q')$6, $(V,q)\supset (V',q')$7 takes values in $(V,q)\supset (V',q')$8 with determinant constrained by the discriminant. Tempered packets are characterized by stability of the sum of characters and transfer to twisted endoscopy; generic non-tempered parameters are obtained from tempered ones by Langlands classification. Concretely, if

$(V,q)\supset (V',q')$9

then the packet $k$0 is defined as the set of Langlands quotients of

$k$1

and is parametrized by characters of the component group $k$2 (Moeglin et al., 2010).

Mœglin and Waldspurger prove the local Gross–Prasad conjecture for all generic $k$3-packets of special orthogonal groups. Their theorem has three parts. First, every induced representation whose Langlands quotient belongs to a generic packet $k$4 or $k$5 is irreducible. Second, if $k$6, then

$k$7

Third, if $k$8, then there exist unique characters $k$9, $$\mathcal{P}(\phi,\phi')=\int_{SO(V',k)\backslash SO(V',\mathbb{A})}\phi(g')\,\overline{\phi'(g')}\,dg',$$0 such that

$$\mathcal{P}(\phi,\phi')=\int_{SO(V',k)\backslash SO(V',\mathbb{A})}\phi(g')\,\overline{\phi'(g')}\,dg',$$1

and every other pair in the product of packets has multiplicity $$\mathcal{P}(\phi,\phi')=\int_{SO(V',k)\backslash SO(V',\mathbb{A})}\phi(g')\,\overline{\phi'(g')}\,dg',$$2 (Moeglin et al., 2010).

The proof reduces generic non-tempered pairs to tempered pairs by compatibility of multiplicity with parabolic induction. For induced representations

$$\mathcal{P}(\phi,\phi')=\int_{SO(V',k)\backslash SO(V',\mathbb{A})}\phi(g')\,\overline{\phi'(g')}\,dg',$$3

with tempered inducing data, Proposition 1.3 gives

$$\mathcal{P}(\phi,\phi')=\int_{SO(V',k)\backslash SO(V',\mathbb{A})}\phi(g')\,\overline{\phi'(g')}\,dg',$$4

The remaining issue is irreducibility of the relevant standard modules. That is handled by the extended cuspidal support and the linkage criterion: if a segment $$\mathcal{P}(\phi,\phi')=\int_{SO(V',k)\backslash SO(V',\mathbb{A})}\phi(g')\,\overline{\phi'(g')}\,dg',$$5 is not linked with $$\mathcal{P}(\phi,\phi')=\int_{SO(V',k)\backslash SO(V',\mathbb{A})}\phi(g')\,\overline{\phi'(g')}\,dg',$$6, then the standard module $$\mathcal{P}(\phi,\phi')=\int_{SO(V',k)\backslash SO(V',\mathbb{A})}\phi(g')\,\overline{\phi'(g')}\,dg',$$7 is irreducible; the global criterion is Theorem 2.13. Muic’s theorem, used in the quasi-split generic case, supplies the key implication that a generic Langlands quotient forces irreducibility of the standard module itself (Moeglin et al., 2010).

4. Refined formulas: squares of periods and local factors

The refined Gross–Prasad philosophy asserts not only non-vanishing but an explicit identity between the square of a period and a central $$\mathcal{P}(\phi,\phi')=\int_{SO(V',k)\backslash SO(V',\mathbb{A})}\phi(g')\,\overline{\phi'(g')}\,dg',$$8-value multiplied by normalized local factors. In the $$\mathcal{P}(\phi,\phi')=\int_{SO(V',k)\backslash SO(V',\mathbb{A})}\phi(g')\,\overline{\phi'(g')}\,dg',$$9 case, Furusawa and Morimoto prove an Ichino–Ikeda type formula for any tempered irreducible cuspidal automorphic representation $\phi\in\pi$0 of $\phi\in\pi$1 with trivial central character. For $\phi\in\pi$2,

$\phi\in\pi$3

where each $\phi\in\pi$4 is built from the local Bessel period $\phi\in\pi$5 and normalized local $\phi\in\pi$6-factors (Furusawa et al., 2022). In this setting, global non-vanishing is equivalent to simultaneous non-vanishing of the central value and all local Bessel models.

For special Bessel periods on $\phi\in\pi$7, Furusawa and Morimoto prove Liu’s refined global Gross–Prasad conjecture under temperedness and discrete-series archimedean hypotheses. If $\phi\in\pi$8 is an irreducible cuspidal tempered automorphic representation of $\phi\in\pi$9 and $SO(V',q')\subset SO(V,q)$00 is the special Bessel period attached to a quadratic extension $SO(V',q')\subset SO(V,q)$01, then for a decomposable cusp form $SO(V',q')\subset SO(V,q)$02,

$SO(V',q')\subset SO(V,q)$03

In degree $SO(V',q')\subset SO(V,q)$04, this yields Boecherer-type formulas for holomorphic Siegel cusp forms via the accidental isomorphism $SO(V',q')\subset SO(V,q)$05 (Furusawa et al., 2016).

Emory formulates a parallel global Gan–Gross–Prasad conjecture for $SO(V',q')\subset SO(V,q)$06, with period

$SO(V',q')\subset SO(V,q)$07

and conjectural identity

$SO(V',q')\subset SO(V,q)$08

When both central characters are trivial and $SO(V',q')\subset SO(V,q)$09, this reduces to the refined orthogonal Gross–Prasad formula on $SO(V',q')\subset SO(V,q)$10 (Emory, 2019). This suggests a structural extension of orthogonal period formulas from special orthogonal groups to their general spin covers.

5. Non-tempered spectra, regularization, and higher-corank developments

The tempered formalism does not exhaust the orthogonal period problem. Qiu treats the nontempered cuspidal spectrum of $SO(V',q')\subset SO(V,q)$11, including Saito–Kurokawa and Soudry packets, where local matrix coefficient integrals are no longer absolutely convergent. For a global Bessel period

$SO(V',q')\subset SO(V,q)$12

the refined formula involves regularized local Bessel functionals $SO(V',q')\subset SO(V,q)$13 and the shifted value $SO(V',q')\subset SO(V,q)$14 rather than a central $SO(V',q')\subset SO(V,q)$15-value. In the $SO(V',q')\subset SO(V,q)$16 nontempered setting, the global period is nonzero if and only if all local Hom spaces are nonzero and $SO(V',q')\subset SO(V,q)$17, and the square of the period factors as an Euler product of regularized local Bessel period integrals (Qiu, 2013).

A recent higher-corank development establishes several non-tempered global Gan–Gross–Prasad cases by relating special periods to special $SO(V',q')\subset SO(V,q)$18-values through theta lifts, tower properties, and Rankin–Selberg integrals. For orthogonal groups, the principal family is $SO(V',q')\subset SO(V,q)$19, where the special Bessel period $SO(V',q')\subset SO(V,q)$20 is shown to be equivalent to non-vanishing and genericity of the first-occurrence theta lift to $SO(V',q')\subset SO(V,q)$21, and hence to the analytic condition that the standard $SO(V',q')\subset SO(V,q)$22-function $SO(V',q')\subset SO(V,q)$23 has a pole at $SO(V',q')\subset SO(V,q)$24. In the parameter language, for a tempered orthogonal $SO(V',q')\subset SO(V,q)$25-parameter $SO(V',q')\subset SO(V,q)$26 for $SO(V',q')\subset SO(V,q)$27 and the non-tempered parameter $SO(V',q')\subset SO(V,q)$28 for $SO(V',q')\subset SO(V,q)$29, one obtains

$SO(V',q')\subset SO(V,q)$30

in the relevant packet (Haan et al., 6 May 2026).

These non-tempered results indicate that orthogonal Gross–Prasad periods persist beyond the generic tempered spectrum, but the analytic framework changes. Regularization of local periods, first-occurrence phenomena in theta towers, and the appearance of shifted special values such as $SO(V',q')\subset SO(V,q)$31 are not accidental technicalities; they are intrinsic features of the non-tempered theory.

6. Archimedean branching, symmetry breaking, and related variants

At the real place, orthogonal Gross–Prasad periods become symmetry breaking problems for pairs such as

$SO(V',q')\subset SO(V,q)$32

Kobayashi and Speh study irreducible unitary representations $SO(V',q')\subset SO(V,q)$33 of $SO(V',q')\subset SO(V,q)$34 and $SO(V',q')\subset SO(V,q)$35 of $SO(V',q')\subset SO(V,q)$36 with infinitesimal character equal to that of the trivial representation. They prove

$SO(V',q')\subset SO(V,q)$37

so the local multiplicity is $SO(V',q')\subset SO(V,q)$38 or $SO(V',q')\subset SO(V,q)$39, and in the tempered range this matches the Gross–Prasad prediction (Kobayashi et al., 2017).

Kobayashi’s later work with Speh formulates the real rank-one branching law for nonsingular integral infinitesimal character in terms of enhanced $SO(V',q')\subset SO(V,q)$40-stable parameters. For irreducible $SO(V',q')\subset SO(V,q)$41 of $SO(V',q')\subset SO(V,q)$42 and $SO(V',q')\subset SO(V,q)$43 of $SO(V',q')\subset SO(V,q)$44, the non-vanishing of

$SO(V',q')\subset SO(V,q)$45

is characterized by equality of signatures, the height condition $SO(V',q')\subset SO(V,q)$46, and interlacing of the corresponding highest-weight data. In the special case of trivial infinitesimal character, any nonzero period is nonzero on the minimal $SO(V',q')\subset SO(V,q)$47-type, and for cohomological representations $SO(V',q')\subset SO(V,q)$48 one obtains $SO(V',q')\subset SO(V,q)$49-distinguished representations and induced nontrivial bilinear forms on $SO(V',q')\subset SO(V,q)$50-cohomology (Kobayashi et al., 2019).

A related but more indirect variant appears in periods of spherical Eisenstein series on rank-one orthogonal groups. For $SO(V',q')\subset SO(V,q)$51 and anisotropic $SO(V',q')\subset SO(V,q)$52, global $SO(V',q')\subset SO(V,q)$53-periods of spherical Eisenstein series factor into Euler products of local integrals, and the dyadic local factors can be evaluated explicitly. The resulting formulas are expressed in terms of zeta factors and local quadratic $SO(V',q')\subset SO(V,q)$54-factors and are stated to be consistent with the Gross–Prasad conjecture (Boavida, 2015). This setting is not the cuspidal Gross–Prasad problem itself, but it exhibits the same period–$SO(V',q')\subset SO(V,q)$55-factor architecture and shows that even-prime local analysis fits the expected orthogonal pattern.

Across these archimedean and rank-one variants, a stable theme emerges: orthogonal Gross–Prasad periods are local restriction functionals whose existence is governed by multiplicity one, precise packet combinatorics, and the geometry of small $SO(V',q')\subset SO(V,q)$56-types or Bessel orbits. A plausible implication is that the classical restriction problem, the modern $SO(V',q')\subset SO(V,q)$57-packet formulation, and refined period identities are not separate theories but different manifestations of one local-global orthogonal period calculus.