$\mathbb{P}{\mathfrak{g}\mathfrak{l}}_2$ is Multiplicity-Free as a $PG{L}_2\times PG{L}_2$-Variety

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: $\dim Ho{m}_{G(F)}(\mathcal{S}(X(F)),\pi )\le 1$ where $\mathcal{S}(X(F))$ is the space of Schwartz functions on $X(F)$. In this thesis we prove that $\mathbb{P}{\mathfrak{g}\mathfrak{l}}_2(F)$ is multiplicity-free as a $PG{L}_2(F)\times PG{L}_2(F)$-variety.