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

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.