Compactly supported cohomology of a tower of graphs and generic representations of $\protect \mathrm{PGL}_{n}$ over a local field

Anis Rajhi

doi:10.5802/crmath.485

Combinatoire, Théorie des représentations

Compactly supported cohomology of a tower of graphs and generic representations of

{PGL}_{n}

over a local field

Anis Rajhi ^1,²

¹ Department of quantitative methods, Higher business school of Tunis, Manouba University, Manouba 2010, Tunisia
² Department of mathematics, Faculty of Sciences and Humanities in Dawadmi, Shaqra University, 11911 Shaqra, Saudi Arabia

Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1133-1149.

Résumé

Let $F$ be a non-archimedean locally compact field and let $G_{n}$ be the group ${PGL}_{n} (F)$ . In this paper we construct a tower ${({\tilde{X}}_{k})}_{k ⩾ 0}$ of graphs fibred over the one-skeleton of the Bruhat–Tits building of $G_{n}$ . We prove that a non-spherical and irreducible generic complex representation of $G_{n}$ can be realized as a quotient of the compactly supported cohomology of the graph ${\tilde{X}}_{k}$ for $k$ large enough. Moreover, when the representation is cuspidal then it has a unique realization in a such model.

Reçu le : 2022-10-31
Révisé le : 2023-02-24
Accepté le : 2023-02-28
Publié le : 2023-10-24

DOI : 10.5802/crmath.485

Classification : 22E50, 20E42

Affiliations des auteurs :

Anis Rajhi ^{1, 2}

¹ Department of quantitative methods, Higher business school of Tunis, Manouba University, Manouba 2010, Tunisia
² Department of mathematics, Faculty of Sciences and Humanities in Dawadmi, Shaqra University, 11911 Shaqra, Saudi Arabia

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Anis Rajhi},
     title = {Compactly supported cohomology of a tower of graphs and generic representations of $\protect \mathrm{PGL}_{n}$ over a local field},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1133--1149},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {361},
     year = {2023},
     doi = {10.5802/crmath.485},
     language = {en},
}

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Anis Rajhi. Compactly supported cohomology of a tower of graphs and generic representations of $\protect \mathrm{PGL}_{n}$ over a local field. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1133-1149. doi : 10.5802/crmath.485. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.485/

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