On generic representations of quasi-split reductive groups over local fields of positive characteristic
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| Format: | Preprint |
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2024
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| _version_ | 1866917852849635328 |
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| author | del Castillo, Héctor Henniart, Guy Lomelí, Luis |
| author_facet | del Castillo, Héctor Henniart, Guy Lomelí, Luis |
| contents | 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 ${\bf G}(F)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_00229 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On generic representations of quasi-split reductive groups over local fields of positive characteristic del Castillo, Héctor Henniart, Guy Lomelí, Luis Representation Theory 22E50 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 ${\bf G}(F)$. |
| title | On generic representations of quasi-split reductive groups over local fields of positive characteristic |
| topic | Representation Theory 22E50 |
| url | https://arxiv.org/abs/2412.00229 |