Supercuspidal L-packets of positive depth and twisted Coxeter elements
Supercuspidal L-packets of positive depth and twisted Coxeter elements
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正深度和扭曲 Coxeter 单元的超尖瓣 L 包
DOI：
10.1515/crelle.2008.046
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发表时间：
2008
期刊：
影响因子：
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通讯作者：
Mark Reeder
中科院分区：
文献类型：
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作者：
Mark Reeder
局部朗兰兹对应是群G（k）在padic域k上的连通约化群G的表示与从k的伽罗瓦（或Weil-Deligne群）到复李群G的某些同态（朗兰兹参数）之间的拓扑联系，复李群G在某种意义上与G对偶，并且编码了k上G的分裂结构。更多关于本地朗兰兹对应的介绍性评论可以在[21]中找到。当G = GL ~ 1时，这种对应关系应归结为局部交换类场论。对于G = GLn，Langlands对应由局部因子唯一确定[24]，并且在[23]和[25]中证明存在。到目前为止，这种对应关系还不完全明确，但在这个方向上已经取得了很大进展;例如，参见[9]，[10]。对于团体以外的GLn或PGLn，理论是先进得多;新的现象出现，所产生的算术方面的差异共轭和稳定的共轭和对偶方面的非平凡monodromy的朗兰兹参数。这意味着一个朗兰兹参数φ不应该只决定一个，而是一个有限的表示集合φ（φ）;这些是标题中的“L-包”。然而，由于局部因子一般没有定义，因此对于一般群，L-包没有精确的特征。目前，人们只能希望定义有限的表示集，并证明它们具有L-包所期望的（或者可能是意料之外的）性质。(See[14，chap. 3]其中一些属性）。因此，人们提出了集合Fi（φ）中表示的局部因子的定义（参见。[4，chap.3]）。本文是[14]的续篇。这两篇论文的目的是验证，在一个明确的和自然的方式，当地朗兰兹对应的最简单的几种非阿贝尔扩张的k，和最简单的几种，
The local Langlands correspondence is a conjectural connection between representations of groups G(k) for connected reductive groups G over a padic field k and certain homomorphisms (Langlands parameters) from the Galois (or Weil-Deligne group) of k into a complex Lie group G which is dual, in a certain sense, to G and which encodes the splitting structure of G over k. More introductory remarks on the local Langlands correspondence can be found in [21]. WhenG = GL1 this correspondence should reduce to local abelian class field theory. For G = GLn, the Langlands correspondence is uniquely determined by local factors [24] and was shown to exist in [23] and [25]. So far this correspondence is not completely explicit, but much progress has been made in this direction; see [9], [10], for example. For groups other than GLn or PGLn, the theory is much less advanced; new phenomena appear, arising on the arithmetic side from the difference between conjugacy and stable conjugacy and on the dual side from nontrivial monodromy of Langlands parameters. This means that a single Langlands parameter φ should determine not just one, but a finite set of representations Π(φ); these are the “L-packets” of the title. However, since local factors have not been defined in general, there is no precise characterization of an L-packet for general groups. One can, at present, only hope to define finite sets of representations Π(φ) attached to Langlands parameters φ, and show that they have properties expected (or perhaps unexpected) of L-packets. (See [14, chap. 3] for some of these properties.) One is thereby proposing a definition of local factors for the representations in the sets Π(φ) (cf. [4, chap.3]). This paper is a sequel to [14]. The aim of both papers is to verify, in an explicit and natural way, the local Langlands correspondence for the simplest kinds of non-abelian extensions of k, and the simplest kinds of