Conductors and newforms for U(1,1)

Let $F$ be a non-Archimedean local field whose residue characteristic is odd. In this paper we develop a theory of newforms for $U(1,1)(F)$, building on previous work on $SL_2(F)$. This theory is analogous to the results of Casselman for $GL_2(F)$ and Jacquet, Piatetski-Shapiro, and Shalika for $GL_n(F)$. To a representation $\pi$ of $U(1,1)(F)$, we attach an integer $c(\pi)$ called the conductor of $\pi$, which depends only on the $L$-packet $\Pi$ containing $\pi$. A newform is a vector in $\pi$ which is essentially fixed by a congruence subgroup of level $c(\pi)$. We show that our newforms are always test vectors for some standard Whittaker functionals, and, in doing so, we give various explicit formulae for newforms.