2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/118085Let $F$ be a finite extension of ${\mathbb{Q}} \_p$. Any dihedral supercuspidal representation of $GL \_2 (K)$ arises from an admissible multiplicative character $ω$ 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 $ω$ 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.Ce document contient 13 pagesRepresentation Theory20G05 ; 20G30 ; 11S31; 11S37text