Local Gan–Gross–Prasad Conjecture

• Local Gan–Gross–Prasad Conjecture is a central prediction in representation theory that links the branching of tempered representations with local epsilon factors.
• The conjecture employs a local relative trace formula that matches geometric and spectral expansions through endoscopic transfer and nilpotent germ computations.
• Its proof unifies p-adic and archimedean techniques, setting a framework for extending branching laws and automorphic period analyses to broader classical groups.

The Local Gan–Gross–Prasad (GGP) Conjecture is a central prediction in the representation theory of classical groups, governing the restriction (branching) of irreducible representations of ambient groups to their subgroups, often in the context of pairs of special orthogonal, unitary, or symplectic groups. The local conjecture explicitly relates distinguished representation pairs in certain Vogan or Arthur packets to arithmetic invariants, notably the sign of local root numbers (epsilon factors) at the central point of Rankin–Selberg or Langlands–Shahidi L-functions. The case for special orthogonal groups SO(n) × SO(n+1) over local fields of characteristic zero was established for tempered representations via local relative trace formulas and geometric–spectral comparison methods, synthesizing results from both p-adic and archimedean settings (Luo, 2020).

1. Formulation of the Local Gan–Gross–Prasad Conjecture

For a local field $F$ of characteristic zero, consider quadratic $F$-spaces $V \supset W$ with $\dim V = n+1$, $\dim W = n$, and the orthogonal complement $W^{\perp}$ split of odd dimension. The ambient group is $G = \mathrm{SO}(W) \times \mathrm{SO}(V)$, and the relevant subgroup is $H = \mathrm{SO}(W) \ltimes N$ where $N$ is the unipotent radical stabilizing a maximal isotropic flag in $W^\perp$. For $(\pi, \sigma)$, irreducible tempered representations of $\mathrm{SO}(W)(F)$ and $\mathrm{SO}(V)(F)$, respectively, and a nondegenerate generic character $\xi: N(F) \to \mathbb{C}^\times$, the multiplicity is defined as

$$m(\pi \otimes \sigma) := \dim \operatorname{Hom}_{H(F)}(\pi \otimes \sigma, \xi) \leq 1.$$

The conjecture asserts a dichotomy: let $\epsilon(1/2, \pi \times \sigma)$ denote the local root number of the Rankin–Selberg or Langlands–Shahidi L-factor at the central point. Then

$$m(\pi \otimes \sigma) = 1 \ \Longleftrightarrow\ \epsilon(1/2, \pi \times \sigma) = +1, \qquad m(\pi \otimes \sigma) = 0 \ \text{otherwise}.$$

This is the "epsilon dichotomy" governing local branching multiplicities.

2. The Relative Trace Formula and Multiplicity Formula

The proof employs a local relative trace formula leveraging Harish–Chandra Schwartz and strongly-cuspidal test functions on $G(F)$. For $f\in S_{\mathrm{cusp}}(G)$,

• The $H\times H$-relative kernel is

$$K_f(x) := \int_{H(F)} f(x^{-1}hx)\,\xi(h)\,dh, \qquad x \in H(F) \setminus G(F).$$

• The trace distribution,

$J(f) := \int_{H(F)\setminus G(F)} K_f(x)\,dx,$

is absolutely convergent on the strongly-cuspidal space.

The trace formula provides two matched expansions:

• Geometric expansion: Via germ expansions at singular orbits,

$$J(f) = \lim_{s\to 0^+} \int_{(G,H)} D^G(x)^{1/2} c_f(x) \Delta_{HG}(x)^{s-1/2}\,dx,$$

where $(G,H)$ runs over semisimple $H(F)$-orbits whose centralizer in $G(F)$ is compact modulo center, $D^G(x)$ is the Weyl discriminant, $c_f(x)$ the regular nilpotent germ of the weighted orbital integral, and $\Delta_{HG}(x) = D^G(x)/D^H(x)^2$ is the relative discriminant.

• Spectral expansion: Via Arthur’s weighted character theory,

$$J(f) = \int_{\mathrm{Temp}(G)} D(\pi) \theta_f(\pi) m(\pi)\,d\pi,$$

where $D(\pi)$ is a formal degree–like measure factor and $m(\pi)$ the branching multiplicity.

The multiplicity $m(\pi)$ for a tempered representation has an explicit formula: $$m(\pi) = \lim_{s \rightarrow 0^+} \int_{(G,H)} D^G(x)^{1/2} c_{\theta_\pi}(x) \Delta_{HG}(x)^{s-1/2} dx,$$ where $\theta_\pi$ is the Harish–Chandra character of $\pi$.

3. Endoscopic Transfer and Matching of Geometric and Spectral Sides

A crucial aspect is matching the geometric and spectral sides via endoscopic transfer:

• Transfer of test functions: Strongly-cuspidal $f$ on $G$ is transferred to matching functions $f'$ on pure inner forms $G'$ so that relative orbital integrals match up to explicit transfer factors. Smooth transfer (the fundamental lemma) is required for both nilpotent orbits and elliptic tori.
• Fundamental lemma and descent: Known fundamental lemmas for Lie algebra orbital integrals (Waldspurger, Shelstad) and descent arguments express regular nilpotent germs in terms of stable germs and endoscopic invariants.
• Function matching and epsilon revelation: The crucial comparison shows $m(\pi) = 1$ precisely when the local root number $\epsilon(1/2, \pi \times \sigma)=+1$, established using explicit computation of nilpotent germs, Shelstad’s formula, and the root number identity.

4. Integration of p-adic and Archimedean Cases

The proof unifies techniques and results across non-archimedean and archimedean fields:

• p-adic case: Waldspurger's trace formula [2010–2013] yields a local relative trace and multiplicity formula for $\mathrm{SO}(n)\times \mathrm{SO}(n+1)$ over $p$-adic fields.
• Archimedean case: Beuzart–Plessis's works provide the analytic and endoscopic ingredients over $\mathbb{R}$, with explicit control over weighted orbital integrals, germ expansions, and spectral expansions.
• In (Luo, 2020), Z. Luo synthesizes these results, verifying that analytic estimates and algebraic-endoscopic properties hold uniformly across all local fields of characteristic zero, resulting in a uniform proof for the tempered local GGP in special orthogonal groups.

5. Canonical Formulas and Normalizations

The analytic apparatus rests on several key formulas and conventions:

• Normalized orbital integrals on the Lie algebra:

$$J_G(X, f) = |D^G(X)|^{1/2} \int_{G(F)/G_X(F)} f(g^{-1} X g)\,dg.$$

• Nilpotent-orbit germs: Shelstad's formula expresses regular nilpotent germs as explicit sums over endoscopic transfer factors indexed by orbits.
• Local root numbers: Defined via Langlands–Shahidi or Rankin–Selberg theory,

$$\epsilon(s, \pi \times \sigma, \psi) \in \{\pm 1\},$$

normalized for holomorphicity at $s=1/2$.

• Character expansion near semisimple $x$:

$$\theta_\pi(x e^Y) \sim |D^G(x e^Y)|^{-1/2} \sum_{\mathcal{O}} c_{\pi, \mathcal{O}}(x)\, \widehat{\mu}_{\mathcal{O}}(Y)$$

as $Y\to 0$.

6. Impact and Extensions

The local Gan–Gross–Prasad conjecture for special orthogonal groups over arbitrary local fields of characteristic zero is now established in the tempered case. The result underpins the broader program relating automorphic periods and central values of L-functions. The trace formula methodology and endoscopic character identities provide a template for extending the conjecture to other classical groups and models (unitary, symplectic–metaplectic, twisted versions) and for handling non-tempered or global cases via analytic continuation, descent, and endoscopy. The epsilon dichotomy is intrinsic to branching multiplicities and has substantial implications for the arithmetic of automorphic forms and special values (Luo, 2020).