On the tempered L-function conjecture

TL;DR

This paper proves Shahidi's tempered L-function conjecture in full generality, establishing key results about the structure and properties of tempered representations of p-adic groups, with implications for the Langlands program.

Contribution

It provides a complete proof of Shahidi's conjecture and confirms the standard modules conjecture for p-adic groups, clarifying the relationship between genericity and irreducibility.

Findings

Proof of Shahidi's tempered L-function conjecture in all cases

Standard modules conjecture for p-adic groups confirmed

Every generic tempered representation is a sub-representation of an induced representation

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 modules 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.