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

Combinatorics, Representation theory

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

Abstract

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.

Received: 2022-10-31
Revised: 2023-02-24
Accepted: 2023-02-28
Published online: 2023-10-24

DOI: 10.5802/crmath.485

Classification: 22E50, 20E42

Author's affiliations:

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

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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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},
     year = {2023},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {361},
     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

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