On the product functor on inner forms of the general linear group over a non-Archimedean local field
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| Format: | Preprint |
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2022
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| _version_ | 1866913234604261376 |
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| author | Chan, Kei Yuen |
| author_facet | Chan, Kei Yuen |
| contents | Let $G_n$ be an inner form of a general linear group over a non-Archimedean field. We fix an arbitrary irreducible representation $σ$ of $G_n$. Lapid-Mínguez give a combinatorial criteria for the irreducibility of parabolic induction when the inducing data is of the form $π\boxtimes σ$ when $π$ is a segment representation. We show that their criteria can be used to define a full subcategory of the category of smooth representation of some $G_m$, on which the parabolic induction functor $τ\mapsto τ\times σ$ is fully-faithful. A key ingredient of our proof for the fully-faithfulness is constructions of indecomposable representations of length 2.
Such result for a special situation has been previously applied in proving the local non-tempered Gan-Gross-Prasad conjecture for non-Archimedean general linear groups. In this article, we apply the fully-faithful result to prove a certain big derivative arising from Jacquet functor satisfies the property that its socle is irreducible and has multiplicity one in the Jordan-Hölder sequence of the big derivative. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2206_10928 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On the product functor on inner forms of the general linear group over a non-Archimedean local field Chan, Kei Yuen Representation Theory Number Theory 11F70, 22E50, 20C08 Let $G_n$ be an inner form of a general linear group over a non-Archimedean field. We fix an arbitrary irreducible representation $σ$ of $G_n$. Lapid-Mínguez give a combinatorial criteria for the irreducibility of parabolic induction when the inducing data is of the form $π\boxtimes σ$ when $π$ is a segment representation. We show that their criteria can be used to define a full subcategory of the category of smooth representation of some $G_m$, on which the parabolic induction functor $τ\mapsto τ\times σ$ is fully-faithful. A key ingredient of our proof for the fully-faithfulness is constructions of indecomposable representations of length 2. Such result for a special situation has been previously applied in proving the local non-tempered Gan-Gross-Prasad conjecture for non-Archimedean general linear groups. In this article, we apply the fully-faithful result to prove a certain big derivative arising from Jacquet functor satisfies the property that its socle is irreducible and has multiplicity one in the Jordan-Hölder sequence of the big derivative. |
| title | On the product functor on inner forms of the general linear group over a non-Archimedean local field |
| topic | Representation Theory Number Theory 11F70, 22E50, 20C08 |
| url | https://arxiv.org/abs/2206.10928 |