Journal of Lie Theory

Let $\cal O$ be the ring of integers of $E$, $E$ being a ramified quadratic extension of a non-archimedean local field $F$ of odd residual characteristic. In this paper, we construct a complete set of irreducible representations $\rho$ of level $n+1$ of the quasi-split unitary group U$(1,1)(\cal O)$ (called primitive representations) such that every irreducible representation of the group has the form $\rho\otimes \chi$ for some character $\chi$ of ${\cal O}^{\times}$. We show that such representations only appear in level $n+1$ when $n$ is even. Our approach is to consider U$(1,1)(\cal O)$ as a generalized special linear group ${\rm SL}^{-1}_*(2,{\cal O})$, i.e., as the group of $2\times 2$ matrices in GL$(2,{\cal O})$ whose coefficients satisfy certain commutation relations involving the nontrivial element $*$ of the Galois group Gal$(E/F)$. Considering $*={\rm id}$ in the construction, we recover the irreducible representations of SL$(2,{\cal O})$. Finally, we explicitly calculate the number and dimensions of the primitive representations so constructed.