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On the Braverman-Kazhdan-Ngo Triples

Dihua Jiang, Zhaolin Li, Guodong Xi

TL;DR

The paper develops the Braverman–Kazhdan–Ngô (BKN) framework for automorphic L-functions by formalizing BKN-triples $(G,G^\vee,\rho)_{\mathfrak{d}}$ and showing that a general $L$-triple can be reduced to a corresponding BKN-triple via a fiber-product construction. It proves that two canonical monoid constructions, Vinberg’s universal L-monoid $\mathcal{M}_\rho$ and the Putcha–Renner monoid $\mathcal{M}^\rho$, are isomorphic in this BKN setting, providing a solid geometric base for global zeta-integral methods. The paper also gives explicit BKN-triples for representative cases (symmetric powers, standard and adjoint representations, symmetric square, and Sp-doubling) and discusses Borel’s conjecture to relate local Langlands data across the fiber-product construction. Together, these results furnish a unified, geometric approach to a broad class of automorphic L-functions and illuminate how local factors and monoid geometry interact within the BKN framework.

Abstract

In the Braverman-Kazhdan proposal and certain refinement of Ngo for automorphic $L$-functions, the reductive group $G$ and the representations $ρ$ of the Langlands dual group $G^\vee$ are taken with certain assumptions. We introduce the notion of the Braverman-Kazhdan-Ngo triples $(G,G^\vee,ρ)$ and show that for general automorphic $L$-functions, it is enough to consider the Braverman-Kazhdan-Ngo triples. We also verify that for a given Braverman-Kazhdan-Ngo triple, the reductive monoid constructed from the Vinberg method and that constructed from the Putcha-Renner method are isomorphic.

On the Braverman-Kazhdan-Ngo Triples

TL;DR

The paper develops the Braverman–Kazhdan–Ngô (BKN) framework for automorphic L-functions by formalizing BKN-triples and showing that a general -triple can be reduced to a corresponding BKN-triple via a fiber-product construction. It proves that two canonical monoid constructions, Vinberg’s universal L-monoid and the Putcha–Renner monoid , are isomorphic in this BKN setting, providing a solid geometric base for global zeta-integral methods. The paper also gives explicit BKN-triples for representative cases (symmetric powers, standard and adjoint representations, symmetric square, and Sp-doubling) and discusses Borel’s conjecture to relate local Langlands data across the fiber-product construction. Together, these results furnish a unified, geometric approach to a broad class of automorphic L-functions and illuminate how local factors and monoid geometry interact within the BKN framework.

Abstract

In the Braverman-Kazhdan proposal and certain refinement of Ngo for automorphic -functions, the reductive group and the representations of the Langlands dual group are taken with certain assumptions. We introduce the notion of the Braverman-Kazhdan-Ngo triples and show that for general automorphic -functions, it is enough to consider the Braverman-Kazhdan-Ngo triples. We also verify that for a given Braverman-Kazhdan-Ngo triple, the reductive monoid constructed from the Vinberg method and that constructed from the Putcha-Renner method are isomorphic.

Paper Structure

This paper contains 15 sections, 18 theorems, 126 equations.

Table of Contents

  1. Introduction
  2. Braverman-Kazhdan-Ngô Triple
  3. BKN-triple via fiber product
  4. Local $L$-factor
  5. $L$-Monoid and BKN-Triple
  6. Construction via the Vinberg method
  7. Construction via the Putcha-Renner method
  8. Proof of Theorem \ref{['thm:monoids']}
  9. Examples
  10. Symmetric powers of ${\mathrm{GL}}_2$
  11. The standard representation of ${\mathrm{GL}}_n$
  12. The adjoint representation of ${\mathrm{GL}}_n$
  13. The symmetric square representation of ${\mathrm{GL}}_n$
  14. The doubling case of ${\mathrm{Sp}}_{2n}$
  15. Borel's Conjecture

Key Result

Theorem 1.4

Let $F$ be a local field of characteristic zero. For any given $L$-triple $(G,G^\vee,\rho)$ with $G$ being $F$-split, there is a BKN-triple $\Delta_{{\mathrm{BKN}}}({\mathfrak {d}})=(G_{\mathfrak {d}},G_{\mathfrak {d}}^\vee,\rho^{\mathfrak {d}})$ such that for any given irreducible admissible repres holds as meromorphic functions in $s\in{\mathbb {C}}$. Note that when $F$ is archimedean, the repre

Theorems & Definitions (26)

  • Definition 1.1: Braverman-Kazhdan-Ngo Triple
  • Conjecture 1.2: Braverman-Kazhdan
  • Definition 1.3: $L$-Triple
  • Theorem 1.4
  • Theorem 1.5
  • Proposition 2.1
  • proof
  • Corollary 2.2
  • Proposition 2.3
  • proof
  • ...and 16 more