Geometric conditions for square-irreducibility of certain representations of the general linear group over a non-archimedean local field

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.