On loop Deligne--Lusztig varieties of Coxeter-type for inner forms of ${\rm GL}_n$
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2019
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| _version_ | 1866912163902259200 |
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| author | Chan, Charlotte Ivanov, Alexander B. |
| author_facet | Chan, Charlotte Ivanov, Alexander B. |
| contents | For a reductive group $G$ over a local non-archimedean field $K$ one can mimic the construction from the classical Deligne--Lusztig theory by using the loop space functor. We study this construction in special the case that $G$ is an inner form of ${\rm GL}_n$ and the loop Deligne--Lusztig variety is of Coxeter type. After simplifying the proof of its representability, our main result is that its $\ell$-adic cohomology realizes many irreducible supercuspidal representations of $G$, notably almost all among those whose L-parameter factors through an unramified elliptic maximal torus of $G$. This gives a purely local, purely geometric and -- in a sense -- quite explicit way to realize special cases of the local Langlands and Jacquet--Langlands correspondences. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1911_03412 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | On loop Deligne--Lusztig varieties of Coxeter-type for inner forms of ${\rm GL}_n$ Chan, Charlotte Ivanov, Alexander B. Algebraic Geometry Number Theory Representation Theory For a reductive group $G$ over a local non-archimedean field $K$ one can mimic the construction from the classical Deligne--Lusztig theory by using the loop space functor. We study this construction in special the case that $G$ is an inner form of ${\rm GL}_n$ and the loop Deligne--Lusztig variety is of Coxeter type. After simplifying the proof of its representability, our main result is that its $\ell$-adic cohomology realizes many irreducible supercuspidal representations of $G$, notably almost all among those whose L-parameter factors through an unramified elliptic maximal torus of $G$. This gives a purely local, purely geometric and -- in a sense -- quite explicit way to realize special cases of the local Langlands and Jacquet--Langlands correspondences. |
| title | On loop Deligne--Lusztig varieties of Coxeter-type for inner forms of ${\rm GL}_n$ |
| topic | Algebraic Geometry Number Theory Representation Theory |
| url | https://arxiv.org/abs/1911.03412 |