On the Braverman-Kazhdan-Ngo Triples

TL;DR

This paper introduces the concept of Braverman-Kazhdan-Ngo triples, demonstrating their sufficiency in automorphic L-functions and establishing isomorphisms between monoids constructed via different methods.

Contribution

It formalizes the notion of Braverman-Kazhdan-Ngo triples and proves their adequacy for automorphic L-functions, also showing the equivalence of monoids from Vinberg and Putcha-Renner constructions.

Findings

Braverman-Kazhdan-Ngo triples are sufficient for automorphic L-functions.

Constructed monoids from different methods are isomorphic.

The notion unifies various approaches in the theory of automorphic L-functions.

Abstract

In the Braverman-Kazhdan proposal and certain refinement of Ngo for automorphic $L$-functions, the reductive group $G$ and the representations $\rho$ of the Langlands dual group $G^\vee$ are taken with certain assumptions. We introduce the notion of the Braverman-Kazhdan-Ngo triples $(G,G^\vee,\rho)$ 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.