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Compactly supported cohomology of a tower of graphs and generic representations of $\protect \mathrm{PGL}_{n}$ over a local field
- Source :
- Comptes Rendus. Mathématique, Vol 361, Iss G7, Pp 1133-1149 (2023)
- Publication Year :
- 2023
- Publisher :
- Académie des sciences, 2023.
-
Abstract
- 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.
- Subjects :
- Mathematics
QA1-939
Subjects
Details
- Language :
- English, French
- ISSN :
- 17783569
- Volume :
- 361
- Issue :
- G7
- Database :
- Directory of Open Access Journals
- Journal :
- Comptes Rendus. Mathématique
- Publication Type :
- Academic Journal
- Accession number :
- edsdoj.3a5044e863b4befaa24a1eeda29ac1f
- Document Type :
- article
- Full Text :
- https://doi.org/10.5802/crmath.485