  Let $\pi$ be an irreducible, complex, smooth representation of $GL_n$ over a
local non-archimedean (skew) field. Assuming $\pi$ has regular Zelevinsky
parameters, we give a geometric necessary and sufficient criterion for the
irreducibility of the parabolic induction of $\pi\otimes\pi$ to $GL_{2n}$. The
latter irreducibility property is the $p$-adic analogue of a special case of
the notion of "real representations" introduced by Leclerc and studied recently
by Kang-Kashiwara-Kim-Oh (in the context of KLR or quantum affine algebras).
Our criterion is in terms of singularities of Schubert varieties of type $A$
and admits a simple combinatorial description. It is also equivalent to a
condition studied by Geiss-Leclerc-Schr\"oer.
