On parabolic induction on inner forms of the general linear group over a non-archimedean local field

Abstract

We give new criteria for the irreducibility of parabolic induction on the general linear group and its inner forms over a local non-archimedean field. In particular, we give a necessary and sufficient condition when the inducing data is of the form π ⊗ σ where π is a ladder representation and σ is an arbitrary irreducible representation. As an application we simplify the proof of the classification of the unitary dual. CONTENTS

Decomposition Rules for the Ring of Representations of Non-Archimedean GLn

“…A classical result due to Bernstein [Ber84] gives a positive answer for that question whenever π 1 , π 2 are unitarizable. The results of Lapid and Mínguez [LM14b,LM16] further deal with the irreducibility question and supply direct combinatorial criteria in some cases. The thesis of Deng Taiwang [Tai16] gives rules for decomposition in the case that π 1 is a segment representation.…”

“…The representationM (m) (respectively, M (m)) has a unique irreducible quotient (respectively, sub-representation), which is isomorphic to L(m). We refer to [LM14b] for a more thorough discussion of the classification with a similar terminology.…”

Models of Representations and Langlands Functoriality

“…Proof. By [12,Lemma 5.17,Proposition 5.20,Lemma 5.21], for any σ ′ ∈ A • E , the representation (π 1 ) (σ ′ ) × · · · × (π k ) (σ ′ ) is irreducible (see §3.1.6 for the notation) and is thus equal to π (σ ′ ) . Therefore, by Lemma 4.2 and Lemma 4.5, we have bc E/F (π) = bc E/F (π 1 ) × · · · × bc E/F (π k ).…”

On two questions concerning representations distinguished by the Galois involution

“…7.0.3. The structure of a representation lying in this class can be described explicitly using [16,Lemma 5.17,Lemma 5.21], which together give a combinatorial criterion to determine exactly when a representation induced from ladders is irreducible. 7.1.…”