2024/11/29 by Guy Henniart, del Castillo, Héctor, Henniart, Guy +2
Mathematics · #Advanced Algebra and Geometry #Finite Group Theory Research #Algebraic Geometry and Number Theory
paper · pdf · doi:10.48550/arxiv.2412.00229
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 π 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 π as above is tempered, then the L-functions attached to π 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 π 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).