On generic representations of quasi-split reductive groups over local fields of positive characteristic

TL;DR

This paper extends key properties of generic representations of quasi-split reductive groups over local fields to positive characteristic, confirming conjectures on $L$-functions, standard modules, and unramified spectra.

Contribution

It proves the tempered $L$-function conjecture, the standard module conjecture, and results on unramified spectra for positive characteristic fields, previously known only in characteristic zero.

Findings

Proof of the tempered $L$-function conjecture for positive characteristic.

Validation of the standard module conjecture in this setting.

Results on the unramified unitary spectrum for split classical groups.

Abstract

Let $F$ be a locally compact non-Archimedean field, and $\bf G$ a connected quasi-split reductive group over $F$. We are interested in complex irreducible smooth generic representations $\pi$ of ${\bf G}(F)$. When $F$ has positive characteristic, we prove important properties which previously were only available for $F$ of characteristic 0. The first one is the tempered $L$-function conjecture of Shahidi, stating that when $\pi$ as above is tempered, then the $L$-functions attached to $\pi$ by the Langlands-Shahidi method have no pole for ${\rm Re}(s)>0$. We also establish the standard module conjecture of Casselman and Shahidi, saying that if $\pi$ is written as the Langlands quotient of a standard module, then it is in fact the full standard module. Finally, for a split classical group $\bf G$ we prove a useful result on the unramified unitary spectrum of ${\bf G}(F)$.