[Paper Review] Local Factors, Reciprocity and Vinberg Monoids

Freydoon Shahidi|arXiv (Cornell University)|Oct 11, 2017

Advanced Algebra and Geometry23 references3 citations

TL;DR

This paper establishes the equality of analytic local factors (L-functions and ε-factors) for arbitrary irreducible representations of $GL_n(bC)$ with their corresponding Artin factors via the local Langlands correspondence, using a generalized multiplicativity axiom based on Schur functors and the Littlewood–Richardson rule. The key contribution is a proof of local–global compatibility for these factors, extending prior results on symmetric and exterior squares to all representations, and linking them to Vinberg monoids and Langlands–Shahidi methods through the structure of $G^\lambda$ and Fourier transforms on monoids.

ABSTRACT

This article addresses the problem of existence of local factors, i.e., the root numbers and L-functions attached to representations of reductive groups over local fields and irreducible finite dimensional representations of their L-groups, as well as their equality with those of Artin factors through the local Langlands correspondence. We conclude the paper with a survey of the theory of monoids of Braverman-Kazhdan, Ngo and Vinberg in generalizing the method of Godement and Jacquet to arbitrary setting and their connections with Langlands-Shahidi method.

Motivation & Objective

• To establish the equality of analytic local factors (L-functions and ε-factors) for arbitrary irreducible finite-dimensional representations of $GL_n(\mathbb{C})$ with their Artin counterparts under the local Langlands correspondence.
• To generalize the multiplicativity axiom used in prior work on $\Lambda^2$ and $\text{Sym}^2$ to arbitrary representations via Schur functors and the Littlewood–Richardson rule.
• To connect the Langlands–Shahidi method with the Braverman–Kazhdan framework by computing the group $G^\lambda$, the group of units of the Vinberg monoid $M^\lambda$, for various representations.
• To show that local coefficients in the Langlands–Shahidi method arise as Fourier transforms of measures defining intertwining operators, linking the theory to Poisson summation and functoriality.

Proposed method

• Uses a generalized multiplicativity axiom (M) based on Schur functors and Young symmetrizers to extend the equality of factors beyond symmetric and exterior squares.
• Applies the Littlewood–Richardson rule to decompose tensor products of representations and verify consistency of factor definitions across all $n$.
• Computes $G^\lambda$, the group of units of the Vinberg monoid $M^\lambda$, for representations $\rho = \text{Sym}^m$ and $\rho = \Lambda^m$ on $GL_n$, particularly for prime $m$, and verifies agreement with known Langlands–Shahidi results.
• Demonstrates that the local coefficients in the Langlands–Shahidi method are Fourier transforms of measures associated with intertwining operators, linking them to Poisson summation.
• Utilizes the framework of Braverman–Kazhdan for $L$-functions via monoids, generalizing the Godement–Jacquet method beyond the standard representation.
• Relies on the local–global compatibility axioms and stability under highly ramified twists to extend results to wild cases, even when representations are not monomial.

Experimental results

Research questions

• RQ1 Does the equality of analytic and Artin local factors hold for all irreducible finite-dimensional representations of $GL_n(\mathbb{C})$ under the local Langlands correspondence?
• RQ2 Can the multiplicativity axiom used for $\Lambda^2$ and $\text{Sym}^2$ be generalized to arbitrary representations using Schur functors and the Littlewood–Richardson rule?
• RQ3 What is the structure of $G^\lambda$, the group of units of the Vinberg monoid $M^\lambda$, for symmetric and exterior powers of $GL_n$?
• RQ4 How do the local coefficients in the Langlands–Shahidi method relate to Fourier transforms of measures defining intertwining operators?
• RQ5 Can the framework of Braverman–Kazhdan, involving Fourier transforms and Poisson summation, be consistently linked to the Langlands–Shahidi method and functoriality?

Key findings

• The equality of analytic and Artin local factors is established for all irreducible finite-dimensional representations of $GL_n(\mathbb{C})$ via the local Langlands correspondence, extending prior results on $\Lambda^2$ and $\text{Sym}^2$.
• The generalized multiplicativity axiom (M), based on Schur functors and the Littlewood–Richardson rule, is sufficient to prove the equality of factors for arbitrary representations.
• For $\rho = \text{Sym}^m$ and $\rho = \Lambda^m$, the group $G^\lambda$ is computed and shown to agree with known Langlands–Shahidi results, particularly when $m$ is prime.
• The local coefficients in the Langlands–Shahidi method are identified as Fourier transforms of measures associated with intertwining operators, providing a new interpretation of these coefficients.
• The structure of $G^\lambda$ is shown to be isomorphic to $\hat{G}^\lambda$, the $L$-group of the group of units of $M^\lambda$, and such pairs $(H,L)$ exist for all finite-dimensional representations of simply connected groups with one-dimensional $G/G_{\text{der}}$.
• The paper confirms that the global functional equation for $L$-functions can be interpreted as a Poisson summation formula via functions of type $L$, linking the theory to functoriality and the Beyond Endoscopy program.