[Paper Review] The functional equation of the Jacquet-Shalika integral representation of the local exterior-square $L$-function

James Cogdell, Nadir Matringe|arXiv (Cornell University)|Jun 7, 2014

Advanced Algebra and Geometry11 references14 citations

TL;DR

This paper establishes the local functional equation for the exterior square L-function of irreducible admissible representations of $GL_n(F)$ for any $n$, extending previous results to the odd-rank case $GL_{2m+1}(F)$ using Bernstein-Zelevinsky derivatives and linear periods. It resolves discrepancies with prior work by Kewat and Raghunathan, confirming the correct global and local functional equations via a purely local approach grounded in the Jacquet-Shalika integral representation.

ABSTRACT

We prove the functional equation of the non archimedean exterior-square L-function of irreducible representations of GL(n), when n is odd.

Motivation & Objective

• To complete the local functional equation for the exterior square L-function in the odd-rank case $GL_{2m+1}(F)$, which was previously unresolved.
• To extend the functional equation to non-generic representations using representations of Whittaker type.
• To correct and clarify discrepancies between the functional equation in this work and that of Kewat and Raghunathan [11], particularly in the odd case.
• To provide a unified local functional equation for all irreducible admissible representations of $GL_n(F)$, enabling future applications in multiplicativity and ramified local theory.

Proposed method

• Uses the Jacquet-Shalika integral representation of the exterior square L-function as the foundational framework.
• Applies Bernstein-Zelevinsky theory of derivatives to analyze the structure of representations and their $L$-functions.
• Extends the theory of linear periods to the odd-rank case, linking them to Shalika periods via earlier results of the second author.
• Employs a purely local approach to derive the functional equation, avoiding reliance on global assumptions.
• Validates the functional equation by deriving its shape from the global functional equation in [9], ensuring consistency.
• Corrects an error in the proof of the $ϵ$-factor being a unit in [14], adapting the argument from Theorem 3.1 to ensure correctness.

Experimental results

Research questions

• RQ1 What is the correct form of the local functional equation for the exterior square $L$-function in the odd-rank case $GL_{2m+1}(F)$?
• RQ2 How can the functional equation be extended to non-generic representations of $GL_n(F)$?
• RQ3 Why does the functional equation in Kewat and Raghunathan [11] differ from the one in Jacquet-Shalika [9], and which is correct?
• RQ4 Can the functional equation be derived purely locally, without relying on global integral representations?
• RQ5 How do the properties of the $ϵ$-factor behave in the odd case, and is it a unit as required for the functional equation?

Key findings

• The local functional equation for the exterior square $L$-function is now fully established for all irreducible admissible representations of $GL_n(F)$, for any $n$, including the odd case $n=2m+1$.
• The functional equation in the odd case is derived via a purely local method based on Bernstein-Zelevinsky derivatives and linear periods, confirming the shape from the global functional equation in [9].
• The functional equation is extended to non-generic representations by using representations of Whittaker type, ensuring broad applicability.
• The paper identifies an error in the global and local functional equations of Kewat and Raghunathan [11], showing their version is incorrect due to mismatched quasi-invariance properties in the integral forms.
• The correct global functional equation, as in [9], includes unipotent integrations on both sides, which are missing in [11], and this difference invalidates their claimed functional equation.
• The correction ensures that the local functional equation derived here is consistent with the global theory and can be used in future applications, such as proving multiplicativity of the $L$-function and $γ$-factor at ramified places.