Shalika models for general linear groups

We define a generalization of Shalika models for $GL_{n+m}(F)$ and prove that they are multiplicity-free, where $F$ is either a non-Archimedean local field or a finite field and $n,m$ are any natural numbers. In particular, we give new proof for the case of $n=m$. We also show that the Bernstein-Zelevinsky product of an irreducible representation of $GL_n(F)$ and the trivial representation of $GL_m(F)$ is multiplicity-free. We relate the two results by a conjecture about twisted parabolic induction of Gelfand pairs.