  Let $F$ be a non-Archimedean local field. Let $G$ be an algebraic group over
$F$. A $G$-variety $X$ defined over $F$ is said to be multiplicity-free if for
any admissible irreducible representation $\pi$ of $G(F)$ the following takes
place: $\dim Hom_{G(F)}(\mathcal{S}(X(F)), \pi) \le 1$ where
$\mathcal{S}(X(F))$ is the space of Schwartz functions on $X(F)$. In this
thesis we prove that $\mathbb{P}\mathfrak{gl}_{2}(F)$ is multiplicity-free as a
$PGL_{2}(F)\times PGL_{2}(F)$-variety.
