Branching rules for principal series representations of unramified U(1,1)

Let $G$ denote the unramified quasi-split unitary group $\mathbb{U}(1,1)(F)$ over a $p$-adic field $F$ with residual characteristic $p\mathrm{\ne }2$. In this paper, we first construct a large family of irreducible representations of the maximal compact subgroup $\mathcal{K}=\mathbb{U}(1,1)({\mathcal{O}}_F)$ of $G$. We then describe the branching rules for all principal series representations of $G$ upon restriction to $\mathcal{K}$ in terms of these representations. The resulting decomposition is multiplicity-free and is characterized by distinct degrees. Finally, we present two important applications of this decomposition that address certain recent open conjectures in the literature. This is the first in a series of two articles in which we provide branching rules for all irreducible smooth representations of the $G$ upon restriction to $\mathcal{K}$.