Epipelagic representations and invariant theory
Epipelagic representations and invariant theory
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远洋表示和不变理论
DOI：
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发表时间：
2013
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通讯作者：
J. Yu
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作者：
Mark Reeder;J. Yu
设G是一个约化p-进群。利用几何不变理论(GIT)中的稳定轨道，我们给出了G(K)的一个新的小深度超尖表示的构造。与已知的方法不同，这种构造对剩余特征p没有任何限制。我们为这些表示构造了适当的朗兰兹参数，对p有一些限制。Git是由分次Moy-Prasad过滤产生的，我们证明了它与Vinberg和Levy分析过其Git的分次Lie代数同构。这导致了Moy-Prasad渗流中的稳定轨道的分类，以及根据绝对根系的Z正则椭圆自同构σ的上层超尖球面表示。转移到G的L群，σ在相应的(野生)朗兰兹参数下产生驯服惯量的映象。对于非分枝群和足够大的p，我们还对具有半稳定轨道的Moy-Prasad渗流进行了分类，解决了G(K)的非退化K型分类这一长期悬而未决的问题。
LetG be a reductive p-adic group. We give a new construction of small-depth “epipelagic” supercuspidal representations ofG(k), using stable orbits in Geometric Invariant Theory (GIT). In contrast to previously known methods, this construction works without any restrictions on the residue characteristic p. We construct appropriate Langlands parameters for these representations, with some restrictions on p. The GIT arises from graded Moy-Prasad filtrations, which we show are isomorphic to the graded Lie algebras whose GIT was analyzed by Vinberg and Levy. This leads to a classification of stable orbits in Moy-Prasad filtrations, as well as epipelagic supercuspidal representations, in terms of Z-regular elliptic automorphisms σ of the absolute root system of G. Transferred to the L-group of G, σ generates the image of tame inertia under the corresponding (wild) Langlands parameter. For unramified groups and sufficiently large p we also classify the Moy-Prasad filtrations which have semi-stable orbits, which solves the long-outstanding problem of classifying non-degenerate K-types for G(k).