Unitary representations of GL(n,K) distinguished by a Galois involution, for K a p-adic field

TL;DR

This paper classifies irreducible unitary representations of GL(n,K) distinguished by GL(n,F) over a p-adic field extension, using Tadic's classification and properties of Steinberg representations.

Contribution

It provides a complete list of distinguished unitary representations of GL(n,K) in terms of distinguished cuspidal representations, extending known criteria.

Findings

Classification of distinguished unitary representations of GL(n,K).

Connection between Steinberg representations and distinguished cuspidal representations.

Explicit criteria for when a representation is distinguished.

Abstract

Let $F$ be a $p$-adic field, and $K$ a quadratic extension of $F$. Using Tadic's classification of the unitary dual of $GL(n,K)$, we give the list of irreducible unitary representations of this group distinguished by $GL(n,F)$, in terms of distinguished discrete series. As it is known that a generalised Steinberg representation $St(k,\rho)$ is distinguished if and only if the cuspidal representation $\rho$ is $\eta^{k-1}$-distinguished, for $\eta$ the character of $F^*$ with kernel the norms of $K^*$, this actually gives a classification of distinguished unitary representations in terms of distinguished cuspidal representations.