%0 Generic %1 chan2022product %A Chan, Kei Yuen %D 2022 %K Indecomposable %T On the product functor on inner forms of general linear group over a non-Archimedean local field %U http://arxiv.org/abs/2206.10928 %X Let $G_n$ be an inner form of a general linear group over a non-Archimedean field. We fix an arbitrary irreducible representation $\sigma$ of $G_n$. Lapid-Mínguez give a combinatorial criteria for the irreducibility of parabolic induction when the inducing data is of the form $\boxtimes \sigma$ when $\pi$ 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 $\tau \sigma$ 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.

@misc{chan2022product, abstract = {Let $G_n$ be an inner form of a general linear group over a non-Archimedean field. We fix an arbitrary irreducible representation $\sigma$ of $G_n$. Lapid-M\'inguez give a combinatorial criteria for the irreducibility of parabolic induction when the inducing data is of the form $\pi \boxtimes \sigma$ when $\pi$ 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 $\tau \mapsto \tau \times \sigma$ 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-H\"older sequence of the big derivative.}, added-at = {2022-06-23T08:41:39.000+0200}, author = {Chan, Kei Yuen}, biburl = {https://www.bibsonomy.org/bibtex/2e9072b949fdfe3dfb5d4012d6b2c9830/dragosf}, description = {On the product functor on inner forms of general linear group over a non-Archimedean local field}, interhash = {d2081e0c299f677cf0eec48fa5f3f5a2}, intrahash = {e9072b949fdfe3dfb5d4012d6b2c9830}, keywords = {Indecomposable}, note = {cite arxiv:2206.10928Comment: 40 pages}, timestamp = {2022-06-23T08:41:39.000+0200}, title = {On the product functor on inner forms of general linear group over a non-Archimedean local field}, url = {http://arxiv.org/abs/2206.10928}, year = 2022 }