Conjectures and results about parabolic induction of representations of $GL_n(F)$

Abstract: In 1980 Zelevinsky introduced commuting varieties whose irreducible components classify complex, irreducible representations of the general linear group over a non-archimedean local field with a given supercuspidal support. We formulate geometric conditions for certain triples of such components and conjecture that these conditions are related to irreducibility of parabolic induction. The conditions are in the spirit of the Geiss--Leclerc--Schr\"oer condition that occurs in the conjectural characterization of $\square$-irreducible representations. We verify some special cases of the new conjecture and check that the geometric and representation-theoretic conditions are compatible in various ways.