On a local conjecture of Jacquet, ladder representations and standard modules

TL;DR

This paper proves that certain ladder representations of GL_n(E) over p-adic fields, which are self-contragredient under Galois conjugation, have the expected distinction properties relative to GL_n(F), confirming a conjecture of Jacquet for a broad class.

Contribution

It establishes the conjecture for a large class of ladder representations by analyzing their standard modules and distinction properties.

Findings

Ladder representations are distinguished with respect to GL_n(F).

The conjecture is reformulated in terms of standard modules.

The result applies to self-contragredient representations under Galois conjugation.

Abstract

Let $E/F$ be a quadratic extension of p-adic fields. We prove that every smooth irreducible ladder representation of the group $GL_n(E)$ which is contragredient to its own Galois conjugate, possesses the expected distinction properties relative to the subgroup $GL_n(F)$. This affirms a conjecture attributed to Jacquet for a large class of representations. Along the way, we prove a reformulation of the conjecture which concerns standard modules in place of irreducible representations.