  Let $F$ be a non-archimedean local field. Given two irreducible
representations $\pi$ and $\sigma$ of $GL_n(F)$ and $GL_m(F)$ we construct an
irreducible representation of $GL_{n+m}(F)$. We conjecture that it occurs as a
subrepresentation of $\pi\times\sigma$, and hence it is the socle of it if
$\pi$ or $\sigma$ is $\square$-irreducible. Some evidence towards this
conjecture is given.
