Conductors and newforms for SL(2)

Joshua M. Lansky and A. Raghuram

Vol. 231 (2007), No. 1, 127–153

DOI: 10.2140/pjm.2007.231.127

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 GL_n(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.

Keywords

conductor, newform, SL(2)

Mathematical Subject Classification 2000

Primary: 22E50

Secondary: 22E35, 11S37, 11S40

Milestones

Received: 30 August 2005

Accepted: 3 August 2006

Published: 1 May 2007