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Abstract
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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.
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Keywords
conductor, newform, SL(2)
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Mathematical Subject Classification
Primary: 22E50
Secondary: 22E35, 11S37, 11S40
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Milestones
Received: 30 August 2005
Accepted: 3 August 2006
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