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 $\pi \otimes \sigma$ where $\pi$ is a ladder representation and $\sigma$ is an arbitrary irreducible representation. As an application we simplify the proof of the classification of the unitary dual.

slightly enhanced version including proof of the conjecture in previous versions (section 5.3)

References in corpus (1)

• Simplicity of heads and socles of tensor products

Cited by in corpus (7)

• Geometric conditions for $\square$-irreducibility of certain representations of the general linear group over a non-archimedean local field
• Classification of standard modules with linear periods
• Conjectures and results about parabolic induction of representations of $G{L}_n(F)$
• Local Rankin--Selberg integrals for Speh representations
• Decomposition rules for the ring of representations of non-Archimedean $G{L}_n$
• Robinson-Schensted-Knuth correspondence in the representation theory of the general linear group over a non-archimedean local field
• Models of representations and Langlands functoriality