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

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.