[Paper Review] Holomorphie des opérateurs d'entrelacement normalisés à l'aide des paramètres d'Arthur

Colette Mœglin|ArXiv.org|Jan 22, 2008

Advanced Algebra and Geometry10 references3 citations

TL;DR

This paper proves that normalized intertwining operators associated with Eisenstein series built from square-integrable automorphic forms and maximal parabolic subgroups of classical groups are holomorphic in a neighborhood of the positive real axis. Using Arthur parameters and a refined inductive approach based on Jacquet modules and compatibility with induction, the authors establish holomorphy by analyzing poles through L-functions and normalization factors, resolving a key technical issue in the theory of automorphic L-functions and Langlands functoriality.

ABSTRACT

In this paper we prove holomorphy for certain intertwining operators arising from the theory of Eisenstein series.

Motivation & Objective

• To establish the holomorphy of normalized intertwining operators for Eisenstein series constructed from square-integrable automorphic forms and maximal parabolic subgroups.
• To resolve the issue of potential poles in the normalization factors of intertwining operators by leveraging Arthur parameter theory.
• To develop a flexible parametrization of Arthur packets that is compatible with standard induction and restriction functors.
• To prove that the normalized intertwining operator $ N(\sigma,\pi,s) = r(\sigma,\psi,s)^{-1} M(\sigma,\pi,s) $ is holomorphic near the positive real axis.
• To extend results from the tempered case (where Langlands and Arthur packets coincide) to the general case using reduction to morphisms with good parity and dominant structures.

Proposed method

• Utilizes Arthur parameters $ \psi $ to parametrize automorphic representations, particularly focusing on those with good parity and dominant refinements $ \psi_{>>} $.
• Applies the theory of Jacquet modules to relate representations in the Arthur packet for $ \psi $ to those for a dominant $ \psi_{>>} $, enabling inductive reduction.
• Employs a refined normalization factor $ r(\sigma,\psi,s) $ derived from Langlands-Shahidi theory, defined via $ L $-functions and standard intertwining operators.
• Reduces the holomorphy problem to analyzing the ratio $ L(St(\rho,a_0)\times\rho, s - B)/L(St(\rho,a_0)\times\rho, s - \tilde{B}) $, showing poles only if $ A_0 = B $, which is ruled out.
• Uses induction on the rank of the group and the structure of the standard intertwining operator $ M(\sigma,\pi,s) $, decomposing it into a product of standard intertwining operators.
• Applies Harish-Chandra's result on holomorphy of standard intertwining operators for discrete series when $ \operatorname{Re}(s) > 0 $, and extends it via normalization and duality.

Experimental results

Research questions

• RQ1 Under what conditions is the normalized intertwining operator $ N(\sigma,\pi,s) $ holomorphic near the positive real axis?
• RQ2 How does the normalization factor $ r(\sigma,\psi,s) $ behave in relation to the standard intertwining operator $ M(\sigma,\pi,s) $, and can it remove poles?
• RQ3 Can the holomorphy of normalized intertwining operators be established in the non-tempered case using Arthur parameter theory?
• RQ4 What role do Jacquet modules and dominant refinements $ \psi_{>>} $ play in reducing the general case to a tractable subcase?
• RQ5 How does the structure of the $ L $-function ratio $ L(\rho \times \rho, s') $ affect the presence of poles in the normalization factor?

Key findings

• The normalized intertwining operator $ N(\sigma,\pi,s) = r(\sigma,\psi,s)^{-1} M(\sigma,\pi,s) $ is holomorphic in a neighborhood of the positive real axis.
• The only potential poles of $ N(\sigma,\pi,s) $ arise from the normalization factor $ r(\sigma,\psi,s)^{-1} $, which is shown to be holomorphic by analyzing the ratio of $ L $-functions.
• The ratio $ L(St(\rho,a_0)\times\rho, s - B)/L(St(\rho,a_0)\times\rho, s - \tilde{B}) $ has a pole only if $ A_0 = B $, but this case is ruled out by the structure of the induced representation.
• The proof relies on an inductive reduction to the case where $ \psi $ has good parity and $ \psi_{>>} $ is dominant and of discrete restriction to the diagonal.
• The construction of Arthur packets via Jacquet modules and the use of flexible parametrization (without rigid choices) enable compatibility with induction and restriction functors.
• The result confirms that the normalization factor $ r(\sigma,\psi,s) $ is sufficient to remove all poles in the standard intertwining operator $ M(\sigma,\pi,s) $ along $ \mathbb{R}_{>0} $, ensuring holomorphy.