On certain representations of the general linear group over a non-archimedean local field

Erez Lapid, Alberto Minguez

Published 2016-05-27Version 1

Let $\pi$ be an irreducible, complex, smooth representation of $GL_n$ over a local non-archimedean (skew) field. We give a simple combinatorial sufficient condition 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 algebras). Our sufficient condition is closely related to singularities of Schubert varieties of type $A$.