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

TL;DR

This paper provides a geometric criterion based on Schubert variety singularities to determine when certain induced representations of $GL_n$ over non-archimedean fields are irreducible, extending the understanding of $p$-adic representation theory.

Contribution

It introduces a new geometric necessary and sufficient condition for the irreducibility of parabolic inductions of representations with regular Zelevinsky parameters, linking algebraic and geometric perspectives.

Findings

Criterion based on Schubert variety singularities for irreducibility

Equivalence with a condition studied by Geiss-Leclerc-Schröer

Connection to real representations in the $p$-adic context

Abstract

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.