Conductors and newforms for U(1,1)

Abstract

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 SL 2 (F). This theory is analogous to the results of Casselman for GL 2 (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.

Conductors and newforms for non-supercuspidal representations of unramified U(2,1)

“…Therefore we need to consider only non-supercuspidal representations. Similar results are obtained by Lansky and Raguram for unramified U(1, 1) and for SL(2)( [8] and [9]), and by Roberts and Schmidt for PGSp(4) ( [14]).…”

“…After Casselman [3], similar results are obtained by Jacquet, Piatetski-Shapiro and Shalika [6] and Reeder [13] for GL n (F ) and by Roberts and Schmidt [14] for PGSp (4). For unitary groups, there is a result by Lansky and Raguram [8]. They computed the dimensions of the spaces of vectors fixed by certain open compact subgroups of unramified U (1,1).…”

Conductors and newforms for SL(2)

“…A companion to this article [Lansky and Raghuram 2004] deals with newforms for the quasisplit unramified unitary group U (1, 1). It would be of interest to generalize these results to other groups, namely, to SL n for higher n and for unitary groups in three variables (for instance the quasisplit unramified unitary group U (2, 1)).…”

Companion forms over totally real fields

“…Thus under the assumption that BC(π) = (ψ, Π) with JL(Π) cuspidal, we have attached a continuous 2-dimensional p-adic representation R 1 (π) to π (note that we have implicitly used the theory of the conductor for U (1, 1) developed in section 5 of [LR04]).…”