Cyclic representations of general linear p-adic groups

Let ${\pi }_1,\dots ,{\pi }_k$ be smooth irreducible representations of $p$-adic general linear groups. We prove that the parabolic induction product ${\pi }_1\times \cdots \times {\pi }_k$ has a unique irreducible quotient whose Langlands parameter is the sum of the parameters of all factors (cyclicity property), assuming that the same property holds for each of the products ${\pi }_i\times {\pi }_j$ ($i<j$), and that for all but at most two representations ${\pi }_i\times {\pi }_i$ remains irreducible (square-irreducibility property). Our technique applies the recently devised Kashiwara-Kim notion of a normal sequence of modules for quiver Hecke algebras. Thus, a general cyclicity problem is reduced to the recent Lapid-M'inguez conjectures on the maximal parabolic case.