Cyclic representations of general linear p-adic groups

Abstract

Let π1,…,πk be smooth irreducible representations of p-adic general linear groups. We prove that the parabolic induction product π1×⋯×πk 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 πi×πj (i<j), and that for all but at most two representations πi×πi 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.