[Paper Review] Non-linear Fourier transforms and the Braverman-Kazhdan conjecture

Tsao-Hsien Chen|arXiv (Cornell University)|Sep 11, 2016

Advanced Algebra and Geometry6 references3 citations

TL;DR

This paper proves the de Rham analogue of the Braverman-Kazhdan conjecture on the acyclicity of gamma sheaves, establishing that convolution with gamma D-modules—viewed as non-linear Fourier transforms—commutes with induction functors and is exact on the category of admissible D-modules on a reductive group. The proof relies on Drinfeld center techniques for Harish-Chandra bimodules and character D-modules, confirming key properties of non-linear Fourier transforms in the D-module setting.

ABSTRACT

In this article we prove a conjecture of Braverman-Kazhdan in \cite{BK} on the acyclicity of gamma sheaves in the de Rham setting. The proof relies on the techniques developed in \cite{BFO} on Drinfeld center of Harish-Chandra bimodules and character D-modules. As an application, we show that the functors of convolution with gamma D-modules, which can be viewed as a version of non-linear Fourier transforms, commute with induction functors and are exact on the category of admissible D-modules on a reductive group.

Motivation & Objective

• To establish the acyclicity of gamma D-modules in the de Rham setting, confirming a key conjecture of Braverman and Kazhdan.
• To verify that the functor of convolution with gamma D-modules behaves as a non-linear Fourier transform, preserving t-structures and commuting with induction.
• To extend the properties of the Fourier-Deligne transform—exactness and compatibility with induction—to the non-linear D-module setting.
• To demonstrate that the convolution functor is exact on the category of admissible D-modules, a central requirement for representation-theoretic applications.
• To link the structure of gamma D-modules to character D-modules and the Drinfeld center of Harish-Chandra bimodules, providing a categorical framework for the results.

Proposed method

• Constructs gamma D-modules ΨG,λ from cocharacters of a maximal torus T, using pushforward via prλ: Gm^r → T and pullback of the exponential D-module on Ga.
• Defines the gamma D-module ΨG,λ as the Weyl-invariant part of the induction of Ψλ from T to G via the Borel subgroup B.
• Applies the Drinfeld center equivalence between Harish-Chandra bimodules and character D-modules to relate the structure of ΨG,λ to monodromic complexes.
• Uses the averaging functor AvU = (πU)* to reduce the acyclicity question to the support of AvU(ΨG,λ) on T = B/U.
• Employs intertwining functors and pro-objects in the derived category of monodromic D-modules to analyze the convolution functor F_G,λ.
• Establishes isomorphisms between convolution with ΨG,λ and the identity functor on monodromic categories, proving exactness via t-exactness of associated functors.

Experimental results

Research questions

• RQ1 Does the convolution functor with gamma D-modules commute with induction functors in the de Rham setting?
• RQ2 Is the functor of convolution with gamma D-modules exact with respect to the perverse t-structure on admissible D-modules?
• RQ3 Is the pushforward of the gamma D-module under the quotient map G → G/U supported on the maximal torus T ⊂ G/U, as conjectured by Braverman and Kazhdan?
• RQ4 Can the non-linear Fourier transform in the D-module setting be understood via the Drinfeld center of Harish-Chandra bimodules?
• RQ5 How do character D-modules and monodromic complexes relate to the structure of gamma D-modules and their convolution properties?

Key findings

• The pushforward of the gamma D-module ΨG,λ under πU: G → G/U is supported on the maximal torus T = B/U, proving the de Rham version of the Braverman-Kazhdan conjecture on acyclicity.
• The convolution functor F_G,λ = (−)*ΨG,λ preserves the category of admissible D-modules and is t-exact with respect to the standard t-structure.
• The functor F_G,λ commutes with induction functors, extending a key property of the classical Fourier-Deligne transform to the non-linear D-module setting.
• The equivalence between the Drinfeld center of Harish-Chandra bimodules and character D-modules underpins the proof, enabling reduction to monodromic complexes.
• The isomorphism AvU(ΨG,λ) ≃ Ψλ and the identity property Ψλ * IX ≃ IX in the pro-category imply that the convolution functor acts as a generalized identity on monodromic categories.
• The proof establishes that HC(F_G,λ(M)) * IY ≃ HC(M) * IY for M in A(G), confirming t-exactness via the conservative and t-exact intertwining functor.