Local Langlands correspondence for reductive p-adic groups

这一提议是关于表象论和数论之间的相互作用，正如一般约化群的推测局部朗兰兹对应（LLC）所预测的那样。有限责任是p进域上局部伽罗瓦群的表示理论与约化群的表示理论之间的预测关系体。本提案的工作目标旨在明确理解LLC。有四个主题：i)扩展提议者早期的工作，引入几何不变量理论来研究p-adic群的表示，特别是最近构建的“上层”表示。ii)离散朗兰兹参数的研究，包括伴随天鹅导体的猜想不等式的证明，小素数下的非典型行为，Yu表示的参数，深度和质量公式的分类。iv)深度零表示和llc的Jordan分解。本提议中的数学植根于两个古老的数学主题：表示理论（对称的研究）和数论（方程的数值解）。虽然这看起来是两个完全不同的数学领域，但《局部朗兰兹通信》预测了它们之间令人惊讶的关系。粗略地说，它说某些无限维对称性应该对应于某些方程，这些方程的解具有有限维对称性的相关集合。该提案的目的首先是发现并明确验证有限责任理论的新的和有趣的例子，其次，利用有限责任理论的预测在数论和表征理论中做出新的发现。

This proposal is about interactions between Representation Theory and Number Theory, as predicted by the conjectural Local Langlands Correspondence (LLC) for general reductive groups. The LLC is a body of predicted relations between the representation theory of local Galois groups and the representation theory of reductive groups over a p-adic field. The goals of the work in this proposal are aimed toward an explicit understanding of the LLC. There are four topics: i) Extending the proposer's earlier work introducing Geometric Invariant Theory to study representations of p-adic groups, in particular the recently-constructed 'epipelagic' representations. ii) Study of discrete Langlands parameters, including proof of a conjectured inequality for adjoint Swan conductors, atypical behavior at small primes, parameters for Yu's representations, classification of depths and mass formulas. iii) Uniqueness results for the LLC. iv) Jordan decomposition of depth zero representations and LLC.The mathematics in this proposal is rooted in two ancient topics of mathematics: Representation Theory (the study of symmetry) and Number Theory (numerical solutions of equations). Though these appear to be two quite different areas of mathematics, the Local Langlands Correspondence predicts surprising relations between them. Roughly speaking, it says that certain kinds of infinite dimensional symmetries should correspond to certain equations whose solutions have a related collection of finite dimensional symmetries. The aims of the proposal are first, to discover and explicitly verify new and interesting examples of the LLC, and second, to use the predictions of the LLC to make new discoveries in Number Theory and Representation Theory.