Conductors and Newforms for 𝑈(1,1)

[摘要] 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 Ï€ 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.

[学科分类] 数学（综合）

[关键词] Conductor;newforms;representations;𝑈(1;1).