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

Abstract

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.

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