On Braverman–Kazhdan–Ngô pairs

Dihua Jiang, Zhaolin Li and Guodong Xi

Vol. 340 (2026), No. 2, 309–337

DOI: 10.2140/pjm.2026.340.309

Abstract

The Braverman–Kazhdan program, later refined by Ngô, aims to understand Langlands $L$-functions attached to a reductive group $G$ and a representation $\rho {:}^{L}G\to {\mathrm{GL}<!--FUNCTION\; APPLICATION-->}_{V_{\rho }}$ of its $L$-group, plus certain additional desiderata. Such pairs $(G,\rho )$ are called Braverman–Kazhdan–Ngô (BKN) pairs. We explain in this paper how it is enough to consider BKN pairs $(G,\rho )$, in order to understand general Langlands $L$-functions. A key tool in the approach of Braverman and Kazhdan is a certain reductive monoid attached to $\rho$. There are two methods of constructing such a reductive monoid in the literature. We prove that the two methods yield the same monoid when $(G,\rho )$ is a BKN pair.

Keywords

reductive monoid, automorphic $L$-function, Braverman–Kazhdan proposal, Borel conjecture

Mathematical Subject Classification

Primary: 11F66, 22E50

Secondary: 11F70

Milestones

Received: 12 December 2024

Revised: 4 August 2025

Accepted: 26 September 2025

Published: 26 November 2025