Generalizations of the Springer correspondence and cuspidal Langlands parameters
Generalizations of the Springer correspondence and cuspidal Langlands parameters
复制标题
施普林格对应和尖朗兰兹参数的概括
DOI：
10.1007/s00229-018-1001-8
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发表时间：
2015
影响因子：
0.6
通讯作者：
M. Solleveld
中科院分区：
文献类型：
--
作者：
A. Aubert;A. Moussaoui;M. Solleveld
设$${\mathcal H}$$ H为任意约化p进群。我们引入了$${\mathcal H}$$ H的增强Langlands参数的cuspidality概念，它推测地将超cuspidality $${\mathcal H}$$ H表示与这些l参数进行双射。我们还定义了一个cuspidal支持映射和增强l -参数的Bernstein分量，类比于Bernstein的p进群表示理论。我们检查了几个著名的还原基，这些类比实际上是精确的。此外，我们还在$${\mathcal H}$$ H的增强l参数空间中发现了一个新的结构，即扭曲扩展商的不相交并的结构。这是ABPS猜想（关于不可约$${\mathcal H}$$ h表示）在局部朗兰兹对应的伽罗瓦侧的一个类似。只是，在伽罗瓦这边，它不再是推测。这些结果将有助于将寻找$${\mathcal H}$$ h表示的局部朗兰兹对应问题简化为$${\mathcal H}$$ h的Levi子群的超尖表示的相应问题。其背后的主要机制来自代数群上的反常束。我们将Lusztig的广义施普林格对应推广到不连通的复约群G，它一方面提供了由G中一个单幂元u组成的对与G中u的中心化子的分量群的不可约表示之间的双射，另一方面提供了若干有限群的一组扭曲群代数的不可约表示。每一个扭曲群代数都包含一个Weyl群的群代数，Weyl群来自于G的中性分量。
Let $${\mathcal H}$$H be any reductive p-adic group. We introduce a notion of cuspidality for enhanced Langlands parameters for $${\mathcal H}$$H, which conjecturally puts supercuspidal $${\mathcal H}$$H-representations in bijection with such L-parameters. We also define a cuspidal support map and Bernstein components for enhanced L-parameters, in analogy with Bernstein’s theory of representations of p-adic groups. We check that for several well-known reductive groups these analogies are actually precise. Furthermore we reveal a new structure in the space of enhanced L-parameters for $${\mathcal H}$$H, that of a disjoint union of twisted extended quotients. This is an analogue of the ABPS conjecture (about irreducible $${\mathcal H}$$H-representations) on the Galois side of the local Langlands correspondence. Only, on the Galois side it is no longer conjectural. These results will be useful to reduce the problem of finding a local Langlands correspondence for $${\mathcal H}$$H-representations to the corresponding problem for supercuspidal representations of Levi subgroups of $${\mathcal H}$$H. The main machinery behind this comes from perverse sheaves on algebraic groups. We extend Lusztig’s generalized Springer correspondence to disconnected complex reductive groups G. It provides a bijection between, on the one hand, pairs consisting of a unipotent element u in G and an irreducible representation of the component group of the centralizer of u in G, and, on the other hand, irreducible representations of a set of twisted group algebras of certain finite groups. Each of these twisted group algebras contains the group algebra of a Weyl group, which comes from the neutral component of G.