Conductors and newforms for SL(2)

Abstract

In this paper we develop a theory of newforms for SL2(F) where F is a nonarchimedean local field whose residue characteristic is odd. This is analogous to results of Casselman for GL2(F) and Jacquet, Piatetski-Shapiro, and Shalika for GLn(F). To a representation π of SL2(F) we attach an integer c(π) that we call the conductor of π. The conductor of π depends only on the L-packet Π containing π. It is shown to be equal to the conductor of a minimal representation of GL2(F) determining the L-packet π. A newform is a vector in π which is essentially fixed by a congruence subgroup of level c(π). For SL2(F) 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.