Note on a conjecture of Braverman-Kazhdan

Given a connected reductive algebraic group G defined over a finite field Fq together with a representation rho(b) : G(b) -> GLN of the dual group of G (in the sense of Deligne-Lusztig), Braverman and Kazhdan [3] defined an exotic Fourier operator on the space of complex valued functions on G(F-q). Under some assumption on rho(b), they gave a conjectural formula for the Fourier kernel which they prove when G = GL(n) for some n. In these notes we give a simple proof of their conjecture for any G without any assumption on rho(b).(c) 2023 Elsevier Inc. All rights reserved.