[Paper Review] On generalized Fourier Transforms for standard L-functions (with an appendix by Wen-Wei Li)

Freydoon Shahidi|arXiv (Cornell University)|Oct 18, 2017

Advanced Algebra and Geometry10 references3 citations

TL;DR

This paper establishes that the Fourier transforms in Braverman-Kazhdan's generalized L-function theory for classical groups match the normalizing factors of the doubling method, and shows both frameworks preserve the unramified (basic) function. The key insight is that a shift of $ s - \frac{1}{2} $ arises naturally from half-density normalization in harmonic analysis, reconciling the differing normalizations in Godement-Jacquet and doubling methods.

ABSTRACT

Any generalization of the method of Godement-Jacquet on principal L-functions for GL(n) to other groups as perceived by Braverman-Kazhdan and Ngo requires a Fourier transform on a space of Schwartz functions. In the case of standard L-functions for classical groups, a theory of this nature was developed by Piatetski-Shapiro and Rallis, called the doubling method. It was later that Braverman and Kazhdan, using an algebro-geometric approach, different from doubling method, introduced a space of Schwartz functions and a Fourier transform, which projected onto those from doubling method. In both methods a normalized intertwining operator played the role of the Fourier transform. The purpose of this paper is to show that the Fourier transform of Braverman-Kazhdan projects onto that of doubling method. In particular, we show that they preserve their corresponding basic functions. The normalizations involved are not the standard ones suggested by Langlands, but rather a singular version of local coefficients of Langlands-Shahidi method. The basic function will require a shift by 1/2 as dictated by doubling construction, reflecting the global theory, and begs explanation when compared with the work of Bouthier-Ngo-Sakellaridis. This matter is further discussed in an appendix by Wen-Wei Li.

Motivation & Objective

• To reconcile the normalizing factors of Braverman-Kazhdan's generalized Fourier transforms with those in the doubling method for standard L-functions of classical groups.
• To show that the unramified (basic) function is preserved under the Braverman-Kazhdan Fourier transform, matching the doubling method's behavior.
• To explain the origin of the $ s - \frac{1}{2} $ shift in the doubling method through geometric and harmonic analytic considerations, particularly using half-densities.
• To clarify the role of the monoid $ M_{\text{ab}} \times G $ and its unit group in defining the Schwartz space and Fourier transform in the Braverman-Kazhdan framework.
• To provide a geometric and analytic explanation for the normalization factor $ c_P |c|^{n+\frac{1}{2}} $, showing it corresponds to a basic half-density on $ X^+(F) $.

Proposed method

• Uses Vinberg's reductive monoids $ M_\rho $ as a geometric replacement for $ M_n $ in Godement-Jacquet theory, with $ M_{\text{ab}} \times G $ as the unit group.
• Defines the Schwartz space as the space of smooth, compactly supported functions on $ M_\rho(F) $, restricted to $ (M_{\text{ab}} \times G)(F) $.
• Applies the $ L^2 $-philosophy from Li [Li] to define 'basic' objects as half-densities: $ c_P |\Xi|^{1/2} $ and $ \mathbb{L}_P^{\text{std}}(\frac{1}{2}) |\Omega|^{1/2} $.
• Relies on explicit calculations of modulus characters and adjoint actions, drawing on [GPSR] and [Sh1], to compare normalizing factors.
• Uses the identity $ \delta_P = |c|^{2n+1} $ on $ (M_{\text{ab}} \times G)(F) $ to relate $ |\Xi| $ and $ |\Omega| $, leading to the half-density normalization.
• Demonstrates that the shift $ n + \frac{1}{2} $ in the normalization factor arises from balancing the half-density $ |\Xi|^{1/2} $ and the modulus character $ \delta_P^{1/2} $.

Experimental results

Research questions

• RQ1 Why does the doubling method require a shift $ s - \frac{1}{2} $ in the standard L-function, while Godement-Jacquet does not?
• RQ2 How are the normalizing factors in Braverman-Kazhdan's Fourier transform related to those in the doubling method?
• RQ3 What is the geometric and analytic justification for the normalization factor $ c_P |c|^{n+\frac{1}{2}} $ in the doubling method?
• RQ4 Why is the function $ \mathbb{L}_P^{\text{std}}(\frac{1}{2}) $ considered 'more basic' than $ \mathbb{L}_P^{\text{std}}(0) $ in the doubling framework?
• RQ5 How does the use of half-densities resolve the discrepancy between the normalizations in Braverman-Kazhdan and doubling methods?

Key findings

• The normalizing factors of the Braverman-Kazhdan Fourier transform and the doubling method's normalized intertwining operators are identical, as shown in Corollary 6.38.
• The Braverman-Kazhdan Fourier transform preserves the unramified (basic) function $ \mathbb{L}_P^{\text{std}}(s) $, as proven in Proposition 6.45.
• The shift $ n + \frac{1}{2} $ in the normalization factor $ c_P |c|^{n+\frac{1}{2}} $ arises naturally from the half-density normalization, where $ \delta_P = |c|^{2n+1} $, so $ \delta_P^{1/2} = |c|^{n+\frac{1}{2}} $.
• The identity $ c_P |\Xi|^{1/2} = \mathbb{L}_P^{\text{std}}(\frac{1}{2}) |\Omega|^{1/2} $ holds on $ X^+(F) $, showing that both sides represent the same basic half-density.
• The basic function $ \mathbb{L}_P^{\text{std}}(\frac{1}{2}) $ is characterized by the zeta integral giving $ L(s + \frac{1}{2}, \pi, \text{std}) $, confirming its centrality in the doubling method.
• The shift $ s - \frac{1}{2} $ is not arbitrary but arises from the interplay of geometry (monoid structure), measure theory (Haar measure), and harmonic analysis (half-densities), as formalized in the $ L^2 $-philosophy of Li [Li].