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Let $\pi$ be an irreducible, complex, smooth representation of $GL_n$ over alocal non-archimedean (skew) field. Assuming $\pi$ has regular Zelevinskyparameters, we give a geometric necessary and sufficient criterion for theirreducibility of the parabolic induction of $\pi\otimes\pi$ to $GL_{2n}$. Thelatter irreducibility property is the $p$-adic analogue of a special case ofthe notion of "real representations" introduced by Leclerc and studied recentlyby 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 acondition studied by Geiss-Leclerc-Schr\"oer.

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