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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 forany admissible irreducible representation $\pi$ of $G(F)$ the following takesplace: $\dim Hom_{G(F)}(\mathcal{S}(X(F)), \pi) \le 1$ where$\mathcal{S}(X(F))$ is the space of Schwartz functions on $X(F)$. In thisthesis we prove that $\mathbb{P}\mathfrak{gl}_{2}(F)$ is multiplicity-free as a$PGL_{2}(F)\times PGL_{2}(F)$-variety.

$\mathbb{P}\mathfrak{gl}_{2}$ is Multiplicity-Free as a $PGL_{2} \times PGL_{2}$-Variety