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 forU (1, 1)(F), building on previous work on SL₂(F). This theory is analogous to the results of Casselman for GL₂(F) and Jacquet, Piatetski-Shapiro, and Shalika for GL_n(F). To a representation π of U(1, 1)(F), we attach an integer c(π) called the conductor of π, which depends only on the L-packet π containing π. A newform is a vector in π which is essentially fixed by a congruence subgroup of level c(π). 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.