On the Braverman-Kazhdan-Ngo Triples

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.