Parabolic induction in the homotopy category of pro-$p$ Iwahori-Hecke modules
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915401993027584 |
|---|---|
| author | Dupré, Nicolas |
| author_facet | Dupré, Nicolas |
| contents | Let $G$ be the group of rational points of a split connected reductive group over a non-archimedean local field of residue characteristic $p$, and let $\mathcal{H}$ denote the pro-$p$ Iwahori-Hecke algebra of $G$ over a field of characteristic $p$. We study the parabolic induction functor for $\mathcal{H}$-modules in terms of the Gorenstein projective model structures introduced by Hovey. Let $\text{Ho}(\mathcal{H})$ denote the associated homotopy category of this model structure. We show that $\text{Ho}(\mathcal{H})$ and its thick subcategory generated by the essential images of finitely many parabolic induction functors are related via a recollement of triangulated categories. We then investigate the isomorphism classes of simple $\mathcal{H}$-modules in $\text{Ho}(\mathcal{H})$ and give a complete classification for simple supersingulars when $G=\mathrm{GL}_n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_12238 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Parabolic induction in the homotopy category of pro-$p$ Iwahori-Hecke modules Dupré, Nicolas Representation Theory Category Theory Number Theory 20C08, 18N40, 18N55 Let $G$ be the group of rational points of a split connected reductive group over a non-archimedean local field of residue characteristic $p$, and let $\mathcal{H}$ denote the pro-$p$ Iwahori-Hecke algebra of $G$ over a field of characteristic $p$. We study the parabolic induction functor for $\mathcal{H}$-modules in terms of the Gorenstein projective model structures introduced by Hovey. Let $\text{Ho}(\mathcal{H})$ denote the associated homotopy category of this model structure. We show that $\text{Ho}(\mathcal{H})$ and its thick subcategory generated by the essential images of finitely many parabolic induction functors are related via a recollement of triangulated categories. We then investigate the isomorphism classes of simple $\mathcal{H}$-modules in $\text{Ho}(\mathcal{H})$ and give a complete classification for simple supersingulars when $G=\mathrm{GL}_n$. |
| title | Parabolic induction in the homotopy category of pro-$p$ Iwahori-Hecke modules |
| topic | Representation Theory Category Theory Number Theory 20C08, 18N40, 18N55 |
| url | https://arxiv.org/abs/2312.12238 |