Characters of p-adic reductive groups in the local Langlands correspondence

该项目属于EPSRC代数研究领域的福尔斯。它符合有影响力和庞大的朗兰兹计划，该计划试图连接和统一数学的大部分，特别是几何，数论，分析和表示论。更具体地说，该项目连接到局部朗兰兹对应，它将（p-adic）约化群的可容许表示与所谓的朗兰兹参数联系起来。我们感兴趣的是这种表示的特征标与它们的Langlands参数之间的关系，为了研究这种关系，我们将考虑Harish-Chandra和Howe的局部特征标展开。这表示了一个约化p-adic群的（光滑可容许）表示的分布特征为幂零轨道积分的傅里叶变换的线性组合。关于特征展开的有效域（通过Waldspurger和DeBacker的作品）已经知道很多，但是线性组合的系数在很大程度上仍然是神秘的。为了掌握这些展开式，我们将尝试将它们的一些系数（即，前导系数）到一定的增长率（“典型维数”），即通过考虑在某些开紧子群的下降链下固定的向量而获得的表示的有限维子空间的维数的增长率。这种增长率受到关注的情况下，真实的还原组和背景下的模p朗兰兹对应，但这条途径的研究仍然是未探索的情况下，复杂的陈述还原p进群。使用真实的还原情况和模p情况为立足点，我们应该能够制定一个令人满意的理论的增长率的维度的不动点子空间的情况下，也还原p-adic群的复表示。我们希望这个理论能阐明局部特征展开，然后我们可以利用这个观点来研究表征的特征与其相关的朗兰兹参数之间的关系，从而有助于阐明局部朗兰兹对应的某些部分。p-adic群的局部朗兰兹对应仍然是不完整的，但其中的一个重要部分，即所谓的幂幺群的分类，现在已经通过Kazhdan、Lusztig、里德、Waldspurger、Opdam、Solleveld和其他人的工作得到了全面的了解。这给了我们一个很好的和有代表性的例子来测试正则维数和朗兰兹参数之间的联系。同样，另一个重要的不变量附加到一个字符的表示是波前集，这是另一种措施的增长。我们想把规范维数与波前集联系起来，这种关系对于真实的群是已知的，但对于p-adic群则不然。特别是，这将是有趣的调查与最近的工作Ciubotaru，梅森布朗，冈田的波前集的unipotent representations.These结果可能会产生影响的表示理论的约化群在当地的领域和数论，在一般框架的朗兰兹计划。

This project falls within the EPSRC Algebra research area. It fits within the influential and massive Langlands program, which attempts to connect and unify large parts of mathematics, in particular geometry, number theory, analysis and representation theory. More specifically the project connects to the local Langlands correspondence, which relates admissible representations of (p-adic) reductive groups to the so-called Langlands parameters. We are interested in the relation between the characters of such representations and their associated Langlands parameters.To study this relation we will consider the local character expansion due to Harish-Chandra and Howe. This expresses the distribution character of a (smooth admissible) representation of a reductive p-adic group as a linear combination of Fourier transforms of nilpotent orbital integrals. Much is known about the domain of validity of the character expansion (by the works of Waldspurger and DeBacker), but the coefficients of the linear combination have largely remained mysterious. To get a handle on these expansions we shall attempt to relate some of their coefficients (i.e., the leading coefficients) to a certain growth rate (the "canonical dimension"), namely the growth rate of the dimensions of the finite-dimensional subspaces of the representation obtained by considering the vectors fixed under a descending chain of certain open compact subgroups. This growth rate received attention in the case of real reductive groups and in the context of the mod p Langlands correspondence, however this avenue of research remains unexplored in the case of complex representations of reductive p-adic groups. Using the real reductive case and the mod p case as footholds, we should be able to develop a satisfactory theory of the growth rate of the dimensions of the fixed point subspaces also in the case of complex representations of reductive p-adic groups. We expect this theory to shed a light on the local character expansion and we can then use this insight to examine the relation between the character of a representation and its associated Langlands parameters, thus helping to elucidate some part of the local Langlands correspondence. The local Langlands correspondence for p-adic groups is still incomplete, but an important part of it, the classification of the so-called unipotent, is now known in full generality by the work of Kazhdan, Lusztig, Reeder, Waldspurger, Opdam, Solleveld and others. This gives us a good and representative source of examples to test the connection between the canonical dimension and the Langlands parameters.In the same vein, another important invariant attached to a character of the representation is the wavefront set, which is another measure of growth. We would like to relate the canonical dimension to the wavefront set, such relations being known for real groups, but not for p-adic groups. In particular, it will be interesting to investigate the relation with the recent work of Ciubotaru, Mason-Brown, and Okada on the wavefront set of unipotent representations.These results are likely to have an impact in the representation theory of reductive groups over local fields and in number theory, in the general framework of the Langlands programme.