The local Gross-Prasad conjecture over $\mathbb{R}$: Epsilon dichotomy

1. Problem Statement

The paper addresses the Local Gross–Prasad (GP) Conjecture for real reductive groups, specifically focusing on the epsilon dichotomy (Part 2) for tempered local L-parameters.

• Context: Let $F = \mathbb{R}$. Consider a pair of non-degenerate quadratic spaces $(W, V)$ over $\mathbb{R}$ with $W \subset V$ and $W^\perp$ split of odd dimension. Let $G = SO(W) \times SO(V)$ and $H = SO(W) \ltimes N$ (where $N$ is a unipotent radical). For an irreducible admissible representation $\pi$ of $G(\mathbb{R})$, the multiplicity is defined as $m(\pi) = \dim \text{Hom}_{H(\mathbb{R})}(\pi, \xi)$, where $\xi$ is a generic character.
• The Conjecture: The GP conjecture predicts that for a generic local L-parameter $\phi$, the sum of multiplicities over the relevant Vogan L-packet is 1 (Part 1), and it identifies which specific representation in the packet has multiplicity 1 based on a character $\chi_\phi$ (Part 2, the epsilon dichotomy).
• Status:
  • Part 1 (Sum of multiplicities) was proved for tempered parameters over $\mathbb{R}$ by the second author (Luo) in [Luo21].
  • Part 2 (Epsilon dichotomy) was known for $p$-adic fields (Waldspurger) and for complex fields, but remained open for tempered parameters over $\mathbb{R}$.
• Goal: Prove that for tempered local L-parameters over $\mathbb{R}$, the unique representation $\pi_\phi$ with $m(\pi_\phi)=1$ satisfies the character relation $\chi_{\pi_\phi} = \chi_\phi$.

2. Methodology

The authors follow the strategy established by Waldspurger for the $p$-adic case ([Wal12b]), adapting it to the archimedean setting. The proof proceeds via a three-step reduction:

A. Reduction to Good Parity

Using results from [MR20] and [Ren24], the authors reduce the general tempered case to parameters of good parity.

• Mechanism: They decompose the standard representation of the L-parameter into "good parity" (O-type) and "bad parity" (Sp-type and GL-type) components.
• Result: Parabolic induction establishes a bijection between the L-packets of the full group and the group restricted to the good parity part, preserving multiplicities and root numbers. This reduces the problem to parameters where the image consists only of O-type irreducible constituents.

B. Endoscopic Reduction

For parameters of good parity, the authors apply ordinary endoscopy to reduce the problem to cases where the component group $S_\phi$ is small (trivial or $\mathbb{Z}/2\mathbb{Z}$).

• Strategy: For an element $s \in S_\phi$, they relate the multiplicity on $G$ to stable geometric multiplicities on endoscopic groups $G^s = SO(V^+) \times SO(V^-) \times SO(W^+) \times SO(W^-)$.
• Key Identity: They aim to prove the identity:
  $$\sum_{\pi \in \Pi_{rel}} \chi_\pi(s) m(\pi) = \chi_\phi(s)$$
  This is transformed into an identity involving stable geometric multiplicities ($m^S_{geom}$).

C. Archimedean Modifications (The Core Technical Contribution)

The $p$-adic proof relies on twisted endoscopy and specific lemmas that do not directly translate to $\mathbb{R}$. The authors introduce two critical modifications:

1. Kottwitz Sign Organization: Unlike the $p$-adic case (where there are few inner forms), $\mathbb{R}$ has many pure inner forms. The authors organize the union of regular semisimple conjugacy classes over pure inner forms by fixing the Kottwitz sign $e(G_\alpha)$. They prove Proposition 3.2.4, describing the union of these classes explicitly in terms of the parameter data.
2. Geometric Multiplicity Formula & Germs: The proof requires recovering the "germ" of a distribution character from the character itself.
   • In the $p$-adic case, this relied on a conjecture by Waldspurger.
   • Over $\mathbb{R}$, the authors prove Lemma 4.2.2, an archimedean analogue of [Wal10b, Lem. 13.4 (ii)], using Rossmann's formula [Ros78] for orbital integrals. This allows them to express the germ $c_\Theta(x)$ as a limit of the quasi-character $\Theta$.
   • They establish Corollary 4.2.3, linking the stable geometric multiplicity to the stable multiplicity.

D. Verification of Basic Cases

After reductions, the problem is reduced to "basic cases" where the L-parameter is either irreducible or a sum of an irreducible and its contragredient (e.g., $SO(2,2) \times SO(2,1)$).

• The authors verify these low-rank cases directly using theta correspondence and low-rank coincidences (e.g., $SO(2,2) \cong SL_2 \times SL_2 / \{\pm 1\}$).
• They utilize see-saw dual pairs and the preservation of genericity under theta correspondence to show the distinguished member is indeed the generic one.

3. Key Contributions

1. Proof of the Epsilon Dichotomy over $\mathbb{R}$: The paper completes the proof of the Local Gross–Prasad conjecture for tempered parameters over the real numbers, resolving the final open case of the conjecture in the archimedean setting.
2. Archimedean Geometric Multiplicity Formula: The authors successfully adapt Waldspurger's geometric multiplicity formula to the real case. This involves:
   • Proving the archimedean analogue of the crucial lemma regarding special values of orbital integrals (Lemma 4.2.2) using Rossmann's formula.
   • Establishing the relationship between stable geometric multiplicities and stable multiplicities (Corollary 5.3.6).
3. Parametrization of Inner Forms: They provide a precise description (Proposition 3.2.4) of the union of regular semisimple conjugacy classes over pure inner forms with a fixed Kottwitz sign, a necessary step for handling the larger set of inner forms in $\mathbb{R}$ compared to $p$-adic fields.
4. Endoscopic Reduction over $\mathbb{R}$: They demonstrate that the endoscopic reduction strategy, previously successful in the $p$-adic case, can be successfully adapted to $\mathbb{R}$ without twisted endoscopy, provided the geometric inputs are correctly handled.

4. Main Results

• Theorem 1.0.2 / Theorem 2.3.2: Over $\mathbb{R}$, Part (2) of the Local Gross–Prasad conjecture holds for tempered local L-parameters. Specifically, if $\pi_\phi$ is the unique representation in the relevant Vogan L-packet with multiplicity 1, then its character $\chi_{\pi_\phi}$ on the component group $S_\phi$ coincides with the explicit character $\chi_\phi$ defined by Gross and Prasad.
• Proposition 3.2.4: A classification of the union of regular semisimple conjugacy classes over pure inner forms with fixed Kottwitz sign.
• Lemma 4.2.2 & Corollary 4.2.3: The derivation of the germ of a distribution character from the character itself in the archimedean setting, enabling the use of the geometric multiplicity formula.
• Proposition 5.3.4: The endoscopic identity relating the sum of multiplicities weighted by characters to the product of stable geometric multiplicities on endoscopic groups.

5. Significance

• Completion of the GP Conjecture: This work, combined with previous results (Luo [Luo21] for Part 1, Chen [Che21] for generic reduction, and Waldspurger for $p$-adic), establishes the full Local Gross–Prasad conjecture for tempered representations over both $\mathbb{R}$ and $\mathbb{C}$.
• Bridge between $p$-adic and Archimedean: The paper demonstrates that the deep structural results of the Langlands program (specifically endoscopy and the Gross–Prasad conjecture) hold uniformly across local fields, despite the significant technical differences in harmonic analysis (orbital integrals, character formulas) between $p$-adic and real fields.
• Methodological Advancement: The successful application of Rossmann's formula to resolve the "germ" issue in the real case provides a template for handling similar geometric multiplicity problems in other archimedean contexts where $p$-adic combinatorial tools fail.
• Foundation for Future Work: The techniques developed here, particularly the handling of Kottwitz signs and stable geometric multiplicities over $\mathbb{R}$, are essential for extending these results to generic (non-tempered) parameters and for further investigations into the global Gross–Prasad conjecture.