Klyachko Models for Ladder Representations

Abstract

We give a new proof for the existence of Klyachko models for unitary representations of ${\mathrm{GL}}_n(F)$ 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 E. Lapid and A. Mínguez [Am. J. Math. 136, No. 1, 111–142 (2014; Zbl 1288.22013)]. 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.