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

Abstract

Let π be an irreducible, complex, smooth representation of GLn over a local non-archimedean (skew) field. Assuming π has regular Zelevinsky parameters, we give a geometric necessary and sufficient criterion for the irreducibility of the parabolic induction of π⊗π to GL2n. 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–Schröer.