Let \(G_n\) be an inner form of a general linear group over a non-Archimedean local field. We fix an arbitrary irreducible representation \(\sigma \) of \(G_n\) . Building on the work of Lapid-Mínguez on the irreducibility of parabolic inductions, we show how to define a full subcategory of the category of smooth representations 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-Hölder sequence of the big derivative.
