Compactly supported cohomology of a tower of graphs and generic representations of $\protect \mathrm{PGL}_{n}$ over a local field
Let $\mathrm{F}$ be a non-archimedean locally compact field and let $\mathrm{G}_{n}$ be the group $\mathrm{PGL}_{n}(\mathrm{F})$. In this paper we construct a tower $(\tilde{\mathrm{X}}_{k})_{k\geqslant 0}$ of graphs fibred over the one-skeleton of the Bruhat–Tits building of $\mathrm{G}_{n}$. We pr...
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Académie des sciences
2023-10-01
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| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.485/ |
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| author | Rajhi, Anis |
| author_facet | Rajhi, Anis |
| author_sort | Rajhi, Anis |
| collection | DOAJ |
| description | Let $\mathrm{F}$ be a non-archimedean locally compact field and let $\mathrm{G}_{n}$ be the group $\mathrm{PGL}_{n}(\mathrm{F})$. In this paper we construct a tower $(\tilde{\mathrm{X}}_{k})_{k\geqslant 0}$ of graphs fibred over the one-skeleton of the Bruhat–Tits building of $\mathrm{G}_{n}$. We prove that a non-spherical and irreducible generic complex representation of $\mathrm{G}_{n}$ can be realized as a quotient of the compactly supported cohomology of the graph $\tilde{\mathrm{X}}_{k}$ for $k$ large enough. Moreover, when the representation is cuspidal then it has a unique realization in a such model. |
| format | Article |
| id | doaj-art-a3a5044e863b4befaa24a1eeda29ac1f |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2023-10-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-a3a5044e863b4befaa24a1eeda29ac1f2025-02-07T11:09:55ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692023-10-01361G71133114910.5802/crmath.48510.5802/crmath.485Compactly supported cohomology of a tower of graphs and generic representations of $\protect \mathrm{PGL}_{n}$ over a local fieldRajhi, Anis0Department of mathematics, Faculty of Sciences and Humanities in Dawadmi, Shaqra University, 11911 Shaqra, Saudi Arabia; Department of quantitative methods, Higher business school of Tunis, Manouba University, Manouba 2010, TunisiaLet $\mathrm{F}$ be a non-archimedean locally compact field and let $\mathrm{G}_{n}$ be the group $\mathrm{PGL}_{n}(\mathrm{F})$. In this paper we construct a tower $(\tilde{\mathrm{X}}_{k})_{k\geqslant 0}$ of graphs fibred over the one-skeleton of the Bruhat–Tits building of $\mathrm{G}_{n}$. We prove that a non-spherical and irreducible generic complex representation of $\mathrm{G}_{n}$ can be realized as a quotient of the compactly supported cohomology of the graph $\tilde{\mathrm{X}}_{k}$ for $k$ large enough. Moreover, when the representation is cuspidal then it has a unique realization in a such model.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.485/ |
| spellingShingle | Rajhi, Anis Compactly supported cohomology of a tower of graphs and generic representations of $\protect \mathrm{PGL}_{n}$ over a local field Comptes Rendus. Mathématique |
| title | Compactly supported cohomology of a tower of graphs and generic representations of $\protect \mathrm{PGL}_{n}$ over a local field |
| title_full | Compactly supported cohomology of a tower of graphs and generic representations of $\protect \mathrm{PGL}_{n}$ over a local field |
| title_fullStr | Compactly supported cohomology of a tower of graphs and generic representations of $\protect \mathrm{PGL}_{n}$ over a local field |
| title_full_unstemmed | Compactly supported cohomology of a tower of graphs and generic representations of $\protect \mathrm{PGL}_{n}$ over a local field |
| title_short | Compactly supported cohomology of a tower of graphs and generic representations of $\protect \mathrm{PGL}_{n}$ over a local field |
| title_sort | compactly supported cohomology of a tower of graphs and generic representations of protect mathrm pgl n over a local field |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.485/ |
| work_keys_str_mv | AT rajhianis compactlysupportedcohomologyofatowerofgraphsandgenericrepresentationsofprotectmathrmpglnoveralocalfield |