Cyclic representations of general linear p-adic groups

TL;DR

This paper proves a cyclicity property for parabolic induction products of p-adic general linear group representations, linking the irreducible quotient's Langlands parameter to the sum of individual parameters under certain assumptions.

Contribution

It establishes a new cyclicity result for induced representations using quiver Hecke algebra techniques, reducing the problem to known conjectures.

Findings

Proves uniqueness of irreducible quotient with summed Langlands parameter.

Reduces cyclicity problem to Lapid-Mínguez conjectures on maximal parabolic cases.

Introduces a novel application of Kashiwara-Kim normal sequences in this context.

Abstract

Let $\pi_1,\ldots,\pi_k$ 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.