Local Rankin-Selberg convolutions for _{}: Explicit conductor formula
Local Rankin-Selberg convolutions for _{}: Explicit conductor formula
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_{} 的局部 Rankin-Selberg 卷积：显式导体公式
DOI：
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发表时间：
1998
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通讯作者：
P. Kutzko
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作者：
C. Bushnell;G. Henniart;P. Kutzko
本文F表示具有q元有限剩余域的非阿基米德局部域。F的离散赋值环记为O，而O是F的加法群的非平凡连续特征标。我们写p为o的最大理想，c（k）为最大整数c，使得p-c <$Ker <$。对于i = 1，2，设ni为正整数，记Gi = GLni（F），设πi为Gi的不可约光滑表示. Jacquet，Piatetskiii-Shapiro和Shalika [14]将L函数L（π1 × π2，s）和局部常数ε（π 1 × π 2，s，ε）附加到数据π1，π2和π 2上，其中s表示复变量。Shahidi [19]的另一种方法将这些对象置于更一般的背景中;无论哪种方式，它们都是局部朗兰兹猜想研究的绝对核心[12]。L函数的形式为L（π1 × π2，s）= P（q−s）−1，其中P（X）∈ C[X ]满足P（0）= 1。另一方面，ε（π1 × π2，s，n）= ε（π1 × π2，0，n）q−f（π1×π2，n）s，对于某个整数f（π1 × π2，n）。事实上，f（π1 × π2，π）= n1 n2 c（π）+ f（π1 × π2），其中f（π1 × π2）与π无关。在[14]中有函数L（π1 × π2，s）的完整描述，同时将局部常数的研究减少到两个πi都是超尖点的情况。给出了当πi为超尖点时f（π1 × π2）的一个显式公式.这个公式（下面的定理6.5）包含了关于表示πi及其相互关系的大量算术信息，就它们作为诱导表示的描述而言。作为这个公式的结果之一，我们得到了f（π1 × π2）的精确的上界和下界，我们现在将解释。为此，我们需要回忆GLn（F）的超尖点表示π的局部常数ε（π，s，ε），在Godement和Jacquet [10]的意义下。这采用ε（π，s，ε）= ε（π，0，ε）q−f（π，ε）s的形式，
In this paper, F denotes a non-Archimedean local field with finite residue field of q elements. The discrete valuation ring of F is denoted o, and ψ is a non-trivial, continuous character of the additive group of F . We write p for the maximal ideal of o and c(ψ) for the largest integer c such that p−c ⊂ Kerψ. For i = 1, 2, let ni be a positive integer, write Gi = GLni(F ), and let πi be an irreducible smooth representation of Gi. To the data π1, π2 and ψ, Jacquet, Piatetskii-Shapiro and Shalika [14] attach an L-function L(π1 × π2, s) and a local constant ε(π1 × π2, s, ψ), where s denotes a complex variable. The alternative approach of Shahidi [19] places these objects in a more general context; either way, they are absolutely central to the study of the local Langlands Conjecture [12]. The L-function has the form L(π1 × π2, s) = P (q−s)−1, where P (X) ∈ C[X ] satisfies P (0) = 1. On the other hand, ε(π1 × π2, s, ψ) = ε(π1 × π2, 0, ψ) q−f(π1×π2,ψ)s, for some integer f(π1 × π2, ψ). In fact, f(π1 × π2, ψ) = n1n2c(ψ) + f(π1 × π2), where f(π1 × π2) is independent of ψ. There is a full description of the function L(π1 × π2, s) in [14] which, at the same time, reduces the study of the local constant to the case where both πi are supercuspidal. The aim of this paper is to give an explicit formula for f(π1 × π2) when the πi are supercuspidal. This formula (Theorem 6.5 below) contains substantial arithmetic information about the representations πi and their relationship to each other, in terms of their description as induced representations [5]. As one of the consequences of this formula, we obtain sharp upper and lower bounds for f(π1 × π2), as we shall now explain. For this purpose, we need to recall the local constant ε(π, s, ψ) of a supercuspidal representation π of GLn(F ), in the sense of Godement and Jacquet [10]. This takes the form ε(π, s, ψ) = ε(π, 0, ψ) q−f(π,ψ)s,