Klyachko models for ladder representations

TL;DR

This paper provides a new local proof for the existence of Klyachko models in unitary representations of GL(n) over non-archimedean fields, classifies distinguished ladder representations, and explores hereditary properties and relations with symplectic distinction.

Contribution

It introduces a purely local approach to Klyachko models, classifies distinguished ladder representations, and analyzes hereditary and induction properties related to these models.

Findings

Classified ladder representations distinguished by Klyachko models.

Proved hereditary property of Klyachko models for induced representations.

Established relations between distinction of parabolic induction and inducing data.

Abstract

We give a new proof of the existence of Klyachko models for unitary representations of ${\rm GL}_{n}(F)$ over a non-archimedean local field $F$. Our methods are purely local and are based on studying distinction within the class of ladder representations introduced by Lapid and Minguez. We classify those ladder representations that are distinguished with respect to Klyachko models. We prove the hereditary property of these models for induced representations from arbitrary finite length representations. Finally, in the other direction and in the context of admissible representations induced from ladder, we study the relation between distinction of the parabolic induction with respect to the symplectic groups and distinction of the inducing data.