Construction of Primitive Representations of U(1,1)(O)

Let 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 of level n + 1 of the quasi-split unitary group U(1, 1)(O) (called primitive representations) such that every irreducible representation of the group has the form for some character of O .We show that such representations only appear in level n + 1 when n is even.Our approach is to consider U(1, 1)(O) as a generalized special linear group SL -1 * (2, O) , i.e., as the group of 2 2 matrices in GL(2, O) whose coefficients satisfy certain commutation relations involving the nontrivial element * of the Galois group Gal(E/F ) .Considering * = id in the construction, we recover the irreducible representations of SL(2, O) .Finally, we explicitly calculate the number and dimensions of the primitive representations so constructed.