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

Abstract

Let $F$ be a locally compact non-Archimedean field, and $\mathbf{G}$ a connected quasi-split reductive group over $F$. We are interested in complex irreducible smooth generic representations $\ddot{I}€$ of $\mathbf{G}(F)$. 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 $\ddot{I}€$ as above is tempered, then the $L$-functions attached to $\ddot{I}€$ by the Langlands-Shahidi method have no pole for $\mathrm{Re}(s)>0$. We also establish the standard module conjecture of Casselman and Shahidi, saying that if $\ddot{I}€$ 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 $\mathbf{G}$ we prove a useful result on the unramified unitary spectrum of $\mathbf{G}(F)$.