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

该项目属于EPSRC代数研究领域。它符合有影响力的大规模朗兰兹计划，该计划试图连接和统一数学的大部分内容，特别是几何、数论、分析和表示理论。更具体地说，该项目与当地的朗兰兹通信有关，该通信将(p-进的)约化群的可接受表示与所谓的朗兰兹参数联系起来。我们感兴趣的是这种表示的特征与其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Harish-Chandra和Howe的局部特征展开。这将一个约化p-ady群的(光滑可容许)表示表示为幂零轨道积分的傅里叶变换的线性组合的分布特征。关于字符扩展的有效性域(Waldspurger和DeBacker的作品)已经知道了很多，但线性组合的系数在很大程度上仍然是个谜。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(“正则维”)联系起来，即通过考虑固定在某些开紧子群的降链下的向量而获得的表示的有限维子空间的维度的增长率。这一增长率在实约化群的情况下和在mod p-朗兰兹对应的背景下受到了关注，然而，在约化p-进群的复表示的情况下，这一研究途径仍然没有被探索。以实约化情形和mod p情形为立足点，我们应该能够发展出一个令人满意的关于不动点子空间的维数增长率的理论，它同样适用于约化p-进群复表示的情形。我们期望这一理论能够揭示局部特征扩展，然后我们可以利用这一洞察力来检验表示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-Add群的局部Langland对应仍然是不完全的，但其中的一个重要部分，即所谓的幂等群的分类，现在已经被Kazhdan，Lusztig，Reeder，Waldspurger，Opam，Solveld等人的工作完全概括地知道了。这为我们提供了一个很好的和有代表性的例子来源来测试规范维度和朗兰兹参数之间的联系。同样，与表示的特征有关的另一个重要不变量是波前集合，这是另一个衡量增长的指标。我们想把正则维与波前集联系起来，这种关系对于实群是已知的，但对于p-进群是未知的。特别地，研究Ciubotaru，Mason-Brown和Okada最近关于幂等表示的波前集的工作的关系将是有趣的。这些结果可能会在局部域上的约化群的表示理论和在朗兰兹计划的一般框架下的数论中产生影响。

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.