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...
Saved in:
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Académie des sciences
2023-10-01
|
| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.485/ |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Summary: | 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. |
|---|---|
| ISSN: | 1778-3569 |