On Certain L-Functions
On Certain L-Functions
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DOI：
10.2307/2374219
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发表时间：
1981-04
影响因子：
1.7
通讯作者：
F. Shahidi
中科院分区：
文献类型：
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作者：
F. Shahidi
我们通过关注问题的一个方面，即嵌入赫克特征值的问题，简要报告了表征的内窥镜分类。 1. G 的问题 我们所说的“特征值”是指自守表示的无分支 Hecke 特征值族。问题是离散谱的特征值是否也是连续谱的特征值。经典群的答案必须是其自同构表示的任何一般分类的一部分。连续谱应在谱定理的意义上狭义地理解。它对应于连续归纳参数是酉的表示。例如，群 SL(2) 的平凡一维自同构表示并不表示嵌入的特征值。这是因为它对应于复数域中非酉点处的一维归纳参数的值。对于一般线性群，缺乏嵌入特征值已经为人所知一段时间了。这是 Jacquet-Shalika [JS] 和 Moeglin-Waldspurger [MW] 分类的结果。对于其他经典群体，这个问题导致了与迹线公式的内窥镜比较相关的有趣的组合问题。我们将考虑 G 是数域 F 上的（简单）拟分裂辛群或特殊正交群的情况。例如，假设 G 是分裂的且等级为 n。然后，最大维度的连续谱由（单一）空闲类字符的 n 元组参数化。是否存在任何 n 元组，其未分支的 Hecke 特征值族与 G 的离散谱中的自同构表示 π 的特征值族相匹配？如果 π 需要有一个全局 Whittaker 模型，那么答案是否定的。这是根据 Cogdell、Kim、Piatetskii-Shapiro 和 Shahidi 2010 年数学学科分类的工作得出的。初级 22E55、22E50；次级20G35、58C40。
We report briefly on an endoscopic classification of representations by focusing on one aspect of the problem, the question of embedded Hecke eigenvalues. 1. The problem for G By “eigenvalue”, we mean the family of unramified Hecke eigenvalues of an automorphic representation. The question is whether there are any eigenvalues for the discrete spectrum that are also eigenvalues for the continuous spectrum. The answer for classical groups has to be part of any general classification of their automorphic representations. The continuous spectrum is to be understood narrowly in the sense of the spectral theorem. It corresponds to representations in which the continuous induction parameter is unitary. For example, the trivial one-dimensional automorphic representation of the group SL(2) does not represent an embedded eigenvalue. This is because it corresponds to a value of the one-dimensional induction parameter at a nonunitary point in the complex domain. For general linear groups, the absence of embedded eigenvalues has been known for some time. It is a consequence of the classification of Jacquet-Shalika [JS] and Moeglin-Waldspurger [MW]. For other classical groups, the problem leads to interesting combinatorial questions related to the endoscopic comparison of trace formulas. We shall consider the case that G is a (simple) quasisplit symplectic or special orthogonal group over a number field F . Suppose for example that G is split and of rank n. The continuous spectrum of maximal dimension is then parametrized by n-tuples of (unitary) idele class characters. Is there any n-tuple whose unramified Hecke eigenvalue family matches that of an automorphic representation π in the discrete spectrum of G? The answer is no if π is required to have a global Whittaker model. This follows from the work of Cogdell, Kim, Piatetskii-Shapiro and Shahidi 2010 Mathematics Subject Classification. Primary 22E55, 22E50; Secondary 20G35, 58C40.