--- title: Orthogonal Gross–Prasad Periods url: https://www.emergentmind.com/topics/orthogonal-gross-prasad-periods type: topic --- # Orthogonal Gross–Prasad Periods 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 [1001.0826], the tempered local conjecture over all local fields of characteristic \(0\) by Luo via a local trace formula [2009.13947], and refined global formulas have been established in several orthogonal Bessel settings [1611.05567], [2205.09503]. ## 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 \(\phi'\in\pi'\), where \(\pi\) and \(\pi'\) are cuspidal automorphic representations of \(SO(V,\mathbb{A})\) and \(SO(V',\mathbb{A})\). In the Gross–Prasad–Ichino–Ikeda formalism, non-vanishing of this period is predicted to be equivalent to non-vanishing of a central value \(L(1/2,\Pi\otimes\Pi')\), subject to local compatibility conditions at every place [1001.0826]. A common misconception is that orthogonal Gross–Prasad periods are only codimension-one restriction integrals. In fact, when \(d' < d-1\), the relevant global periods are frequently Bessel periods. In the \((\mathrm{SO}(5),\mathrm{SO}(2))\) setting, Furusawa and Morimoto realize the orthogonal period on an inner form \(\mathrm{SO}(V)\simeq GD/Z_D\) as a Bessel period over the subgroup \(R_\xi=T_\xi N_D\), \[ \mathcal{B}_{E,\Lambda,\xi}(f) =\int_{\mathbb{A}^\times R_\xi(F)\backslash R_\xi(\mathbb{A})} f(r)\,\chi_{\xi,\Lambda}(r)^{-1}\,dr, \] where \(T_\xi(F)\cong E^\times\) and \(\chi_{\xi,\Lambda}(tn)=\Lambda(t)\psi_\xi(n)\) [2205.09503]. In the higher-rank special Bessel setting for \(\mathrm{SO}(2n+1)\times \mathrm{SO}(2)\), Furusawa and Morimoto define \[ B_\lambda(\varphi) =\int_{R_\lambda(F)\backslash R_\lambda(\mathbb{A})}\varphi(r)\,\chi_\lambda(r)^{-1}\,dr, \] and call it special when the representation of \(\mathrm{SO}(2)\) is trivial [1611.05567]. 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 \(F\) be a non-archimedean local field of characteristic \(0\), let \((V,q)\) and \((V',q')\) have dimensions \(d>d'\) of opposite parity, and write \(G=SO(V,q)\), \(G'=SO(V',q')\). When \(d'=d-1\), the Gross–Prasad multiplicity is \[ m(\sigma,\sigma')=\dim \operatorname{Hom}_{G'(F)}(\sigma,\sigma'). \] For \(d'b_1>\cdots>b_t>0, \] then the packet \(\Pi_G(\varphi)\) is defined as the set of Langlands quotients of \[ \mathrm{Ind}_P^G\bigl(\pi(\varphi_1)|\det|^{b_1}\otimes\cdots\otimes\pi(\varphi_t)|\det|^{b_t}\otimes \sigma_0\bigr), \qquad \sigma_0\in\Pi_{G_0}(\varphi_0), \] and is parametrized by characters of the component group \(S(\varphi)/S(\varphi)^0\) [1001.0826]. Mœglin and Waldspurger prove the local Gross–Prasad conjecture for all generic \(L\)-packets of special orthogonal groups. Their theorem has three parts. First, every induced representation whose Langlands quotient belongs to a generic packet \(\Pi_G(\varphi)\) or \(\Pi_{G'}(\varphi')\) is irreducible. Second, if \(E(\varphi,\varphi')=-(G,G')\), then \[ m(\sigma,\sigma')=0 \quad \text{for all } \sigma\in\Pi_G(\varphi),\ \sigma'\in\Pi_{G'}(\varphi'). \] Third, if \(E(\varphi,\varphi')=(G,G')\), then there exist unique characters \(\epsilon\in E_G(\varphi)\), \(\epsilon'\in E_{G'}(\varphi')\) such that \[ m\bigl(\sigma(\varphi,\epsilon),\sigma'(\varphi',\epsilon')\bigr)=1, \] and every other pair in the product of packets has multiplicity \(0\) [1001.0826]. The proof reduces generic non-tempered pairs to tempered pairs by compatibility of multiplicity with parabolic induction. For induced representations \[ \sigma=\pi_1|\cdot|^{b_1}\times\cdots\times \pi_t|\cdot|^{b_t}\times \sigma_0, \qquad \sigma'=\pi'_1|\cdot|^{b'_1}\times\cdots\times \pi'_{t'}|\cdot|^{b'_{t'}}\times \sigma'_0, \] with tempered inducing data, Proposition 1.3 gives \[ m(\sigma,\check{\sigma}')=m(\sigma_0,\check{\sigma}'_0). \] 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 \((\rho,e,f)\) is not linked with \(\mathrm{Jord}(\pi)\), then the standard module \(\langle e,f\rangle_\rho\times \pi\) 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 [1001.0826]. ## 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 \(L\)-value multiplied by normalized local factors. In the \((\mathrm{SO}(5),\mathrm{SO}(2))\) case, Furusawa and Morimoto prove an Ichino–Ikeda type formula for any tempered irreducible cuspidal automorphic representation \(\pi\) of \(GD(\mathbb{A})\) with trivial central character. For \(\varphi=\otimes_v\varphi_v\in\mathcal V_\pi\), \[ \frac{\big|\mathcal{B}_{E,\Lambda,\xi}(\varphi)\big|^2}{(\varphi,\varphi)_\pi} = 2^{-c(\pi)}\,C_\xi\,\frac{\zeta_F(2)\,\zeta_F(4)} {L(1,\pi,\mathrm{Ad})\,L(1,\chi_E)} \, \frac{L\bigl(\tfrac12,\pi\times AI(\Lambda)\bigr)} {\prod_v \alpha_v(\varphi_v)}, \] where each \(\alpha_v\) is built from the local Bessel period \(\mathcal{B}_{\xi,\Lambda,v}\) and normalized local \(L\)-factors [2205.09503]. 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 \(\mathrm{SO}(2n+1)\times \mathrm{SO}(2)\), Furusawa and Morimoto prove Liu’s refined global Gross–Prasad conjecture under temperedness and discrete-series archimedean hypotheses. If \(\pi\) is an irreducible cuspidal tempered automorphic representation of \(\mathrm{SO}(2n+1)(\mathbb{A})\) and \(B_\lambda\) is the special Bessel period attached to a quadratic extension \(E/F\), then for a decomposable cusp form \(\varphi=\otimes_v\varphi_v\), \[ \frac{|B_\lambda(\varphi)|^2}{(\varphi,\varphi)} = 2^{-1} C_1 \cdot \prod_{j=1}^n \zeta_F(2j) \cdot \frac{L(1/2,\pi)\,L(1/2,\pi\otimes\chi_E)} {L(1,\pi,\mathrm{Ad})\,L(1,\chi_E)} \cdot \prod_v a_v^\circ(\varphi_v). \] In degree \(2\), this yields Boecherer-type formulas for holomorphic Siegel cusp forms via the accidental isomorphism \(\mathrm{SO}(5)\simeq \mathrm{PGSp}_4\) [1611.05567]. Emory formulates a parallel global Gan–Gross–Prasad conjecture for \(GSpin_{n+1}\times GSpin_n\), with period \[ \mathcal{P}_\omega(\phi,f) = \int_{Z_n^\circ(\mathbb A)\,GSpin_n(F)\backslash GSpin_n(\mathbb A)} \phi(g)\,f(g)\,\omega(N(g))^{-1}\,dg, \] and conjectural identity \[ \big|\mathcal{P}_\omega(\phi,f)\big|^2 = \frac{\Delta_{SO_{n+1}}}{2^\beta}\, \frac{L\bigl(1/2,\pi_{n+1}\times\pi_n\otimes\omega^{-1}\bigr)} {L(1,\pi_{n+1},\Ad)\,L(1,\pi_n,\Ad)} \, \prod_v \alpha_{\omega_v}(\phi_v,f_v). \] When both central characters are trivial and \(\omega=1\), this reduces to the refined orthogonal Gross–Prasad formula on \(SO_{n+1}\times SO_n\) [1901.01746]. 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 \(\mathrm{SO}_5\), including Saito–Kurokawa and Soudry packets, where local matrix coefficient integrals are no longer absolutely convergent. For a global Bessel period \[ \mathcal P(f)=\int_{[U\rtimes \mathrm{SO}(V_0)]} f(uh)\,\chi(h)\,\psi_{v_0}(u)\,du\,dh, \] the refined formula involves regularized local Bessel functionals \(\mathcal P_v^\sharp\) and the shifted value \(L(0,\Pi,\chi)\) rather than a central \(L(1/2)\)-value. In the \(\mathrm{SO}_5\) nontempered setting, the global period is nonzero if and only if all local Hom spaces are nonzero and \(L(0,\Pi,\chi)\neq 0\), and the square of the period factors as an Euler product of regularized local Bessel period integrals [1312.5793]. A recent higher-corank development establishes several non-tempered global Gan–Gross–Prasad cases by relating special periods to special \(L\)-values through theta lifts, tower properties, and Rankin–Selberg integrals. For orthogonal groups, the principal family is \((SO_{2n},SO_3)\), where the special Bessel period \(\mathcal B_{n-1,\psi}^0(\pi)\) is shown to be equivalent to non-vanishing and genericity of the first-occurrence theta lift to \(Sp_{2n-2}\), and hence to the analytic condition that the standard \(L\)-function \(L(s,\pi)\) has a pole at \(s=1\). In the parameter language, for a tempered orthogonal \(A\)-parameter \(M\) for \(SO_{2n}\) and the non-tempered parameter \(N=[1]\) for \(SO_3\), one obtains \[ \mathcal B_{n-1,\psi}^0(\pi)\neq 0 \quad\Longleftrightarrow\quad L(0,M,N)\neq 0 \] in the relevant packet [2605.04389]. 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 \(L(0,\Pi,\chi)\) 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 \[ (G,H)=(SO(n+1,1),SO(n,1)). \] Kobayashi and Speh study irreducible unitary representations \(\Pi_{i,\delta}\) of \(SO(n+1,1)\) and \(\pi_{j,\varepsilon}\) of \(SO(n,1)\) with infinitesimal character equal to that of the trivial representation. They prove \[ \dim \operatorname{Hom}_{SO(n,1)}(\Pi_{i,\delta}|_{SO(n,1)},\pi_{j,\varepsilon}) = \begin{cases} 1 & \text{if }\delta=\varepsilon\text{ and }j\in\{i-1,i\},\\ 0 & \text{otherwise,} \end{cases} \] so the local multiplicity is \(0\) or \(1\), and in the tempered range this matches the Gross–Prasad prediction [1702.00263]. Kobayashi’s later work with Speh formulates the real rank-one branching law for nonsingular integral infinitesimal character in terms of enhanced \(\theta\)-stable parameters. For irreducible \(\Pi\) of \(SO(n+1,1)\) and \(\pi\) of \(SO(n,1)\), the non-vanishing of \[ \operatorname{Hom}_{SO(n,1)}(\Pi|_{SO(n,1)},\pi) \] is characterized by equality of signatures, the height condition \(h(\pi)\in\{h(\Pi),h(\Pi)-1\}\), and interlacing of the corresponding highest-weight data. In the special case of trivial infinitesimal character, any nonzero period is nonzero on the minimal \(K\)-type, and for cohomological representations \(A_{\mathfrak q_i}(\lambda)\) one obtains \(SO(n+1-i,1)\)-distinguished representations and induced nontrivial bilinear forms on \((\mathfrak g,K)\)-cohomology [1907.05890]. A related but more indirect variant appears in periods of spherical Eisenstein series on rank-one orthogonal groups. For \(G=O(n+3)\) and anisotropic \(H=O(n+2)\), global \(H\)-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 \(L\)-factors and are stated to be consistent with the Gross–Prasad conjecture [1502.00847]. This setting is not the cuspidal Gross–Prasad problem itself, but it exhibits the same period–\(L\)-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 \(K\)-types or Bessel orbits. A plausible implication is that the classical restriction problem, the modern \(L\)-packet formulation, and refined period identities are not separate theories but different manifestations of one local-global orthogonal period calculus. Source: https://www.emergentmind.com/topics/orthogonal-gross-prasad-periods