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

该项目属于EPSRC代数研究领域。它符合有影响力的、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、分析和表示理论。更具体地说，该项目连接到局部朗兰兹对应，它将（p进）约化群的可接受表示与所谓的朗兰兹参数联系起来。我们感兴趣的是这些表征的特征和它们相关的朗兰兹参数之间的关系。为了研究这种关系，我们将考虑由于Harish-Chandra和Howe引起的局部字符扩展。用幂零轨道积分的傅里叶变换的线性组合来表示一个约化p进群的（光滑可容许）表示的分布特征。关于字符扩展的有效性域（由Waldspurger和DeBacker的作品），我们知道很多，但线性组合的系数在很大程度上仍然是神秘的。为了处理这些展开式，我们将尝试将它们的一些系数（即前导系数）与一定的增长率（“规范维数”）联系起来，即通过考虑在某些开紧子群的下降链下固定的向量而得到的表示的有限维子空间的维数的增长率。这种增长率在实约化群和模p朗兰兹对应的情况下受到了关注，然而，在约化p进群的复杂表示情况下，这一研究途径仍未得到探索。以实约化情形和模p情形为基础，我们应该能够在约p进群的复表示情况下，发展出一个令人满意的不动点子空间维数增长率的理论。我们希望这一理论能够揭示局部特征扩展，然后我们可以利用这一见解来检查表征特征与其相关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进群的局部朗兰兹对应仍然不完整，但它的一个重要部分，即所谓的单能性的分类，现在已经被Kazhdan、Lusztig、Reeder、Waldspurger、Opdam、Solleveld和其他人的工作所全面地了解。这为我们提供了一个很好的、有代表性的例子来测试规范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表征的特征是波前集，这是增长的另一个度量。我们想把规范维数与波前集联系起来，这种关系对于实数群是已知的，但对于p进群却不知道。特别是，研究与最近Ciubotaru， Mason-Brown和Okada关于波前集的无幂表示的关系将是有趣的。这些结果很可能对局部域上约化群的表示理论和朗兰兹方案的一般框架中的数论产生影响。

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.