Conductors and newforms for U(1,1)

Abstract

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 $S{L}_2(F)$. This theory is analogous to the results of Casselman for $G{L}_2(F)$ and Jacquet, Piatetski-Shapiro, and Shalika for $G{L}_n(F)$. 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.

25 pages