Distinguished dihedral representations of $GL(2)$ over a p-adic field

Overview & Synopsis

Let $F$ be a finite extension of ${\mathbb{Q}} _p$. Any dihedral supercuspidal representation of $GL _2 (K)$ arises from an admissible multiplicative character $\omega$ of a quadratic extension $L$ of $K$. We show that such a representation is distinguished for $GL _2 (F)$ if and only if $L$ biquadratic over $F$ and $\omega$ restricted to invertibles of one of the two other quadratic extensions of $F$ in $L$ is trivial. We then observe a similar statement for the principal series and we study all dihedral representations.