Cyclic representations of general linear p-adic groups

Abstract

Let $π_1,\ldots,π_k$ be smooth irreducible representations of $p$-adic general linear groups. We prove that the parabolic induction product $π_1\times\cdots\times π_k$ 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 $π_i\times π_j$ ($i<j$), and that for all but at most two representations $π_i\times π_i$ 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-Mínguez conjectures on the maximal parabolic case.