On the tempered $L$-functions conjecture

Abstract

We give a general proof of Shahidi's tempered L -function conjecture, which has previously been known in all but one case. One of the consequences is the standard module conjecture for $p$-adic groups, which means that the Langlands quotient of a standard module is generic if and only if the standard module is irreducible and the inducing data generic. We have also included the result that every generic tempered representation of a $p$-adic group is a sub-representation of a representation parabolically induced from a generic supercuspidal representation with a non-negative real central character.