Distinction problems by Iwahori-Hecke algebras

Iwahori-Hecke 代数的区分问题

• 批准号: 2283617
• 金额: --
• 依托单位: University of Oxford
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2019
• 资助国家: 英国
• 起止时间: 2019 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2283617
• 关键词: Distinction; problems; Iwahori; Hecke; algebras

This project falls within the ESPRC Algebra research area. The general subject area of this project is the representation theory of p-adic reductive groups, in the general framework of the local Langlands programme. Motivated by applications to automorphic forms, the aim of the project is to study certain distinction questions for smooth representations of p-adic groups or finite groups of Lie type. Consider a reductive p-adic or a finite group of Lie type G and a closed subgroup H of G and two smooth (complex) irreducible representations \pi of G and \sigma of H. The most basic case is when \sigma is the trivial representation. The question we are interested in is when Hom_H(\pi,\sigma) is non-zero and what its dimension is in the case that it is non-zero. In the most interesting cases, this dimension is equal to 0 or 1. This type of problem has a long history. A very classical example is when H is a maximal compact open subgroup of G, for example G=GL(n,Q_p) and H=GL(n,Z_p). The representations distinguished by this H are called (H-)spherical and the fact that the multiplicity space is at most one-dimensional follows from the fact that the spherical Hecke algebra H(G,H) of compactly supported functions of G which are H-biinvariant is abelian. Another famous example is the case of Whittaker (or generic) representations, in which case, H is the unipotent radical of a Borel subgroup (when G is quasisplit) and \sigma is a nondegenerate character of H [CS80].One particularly interesting case to consider is when \pi is a smooth representation of G with Iwahori fixed vectors. It turns out that the category of such representations is equivalent to the category of modules over Iwahori-Hecke algebras H(G,I) of compactly supported I-biinvariant functions. As a consequence, we can study such representations from the point of view of Iwahori-Hecke algebras. The main challenge that comes with this approach is transferring the questions we are studying about the smooth representations to the equivalent question in the Iwahori-Hecke algebras setting. The novelty in this research methodology is the emphasis of this algebraic approach from the point of view of Iwahori-Hecke algebras.There are multiple objectives that we could aim to reach in this project. For instance, one could try to generalise the results of [CS16] who consider representations with Whittaker models. In [CS16], G is a Chevalley group over a p-adic field, H is a unipotent radical of a Borel subgroup of G and \sigma is a non-degenerate character of H. The equivalent problem in the setting of the Iwahori Hecke algebra is to determine the simple modules whose restriction to the finite Hecke algebra contains the Steinberg module. Another objective could be to consider the example with G = GL(2n,\Q_p), H = SP(2n,\Q_p) and \sigma the trivial character, a case studied before via different methods in [OS07]. As a related toy example, one would have to consider the irreducible representations of the symmetric group whose restriction to the hyperoctahedral subgroup contains the trivial representation. More interestingly, insight should be obtained by considering the similar restriction problem for finite groups of Lie type over F_q. The most ambitious goal would be to find an Iwahori-Hecke algebra common framework which applies to a vast class of distinction problems, which include as particular cases the examples mentioned above.

这个项目属于ESPRC代数研究领域的福尔斯。该项目的一般主题领域是p进还原群的表示理论，在当地朗兰兹方案的一般框架。受自守形式应用的启发，该项目的目的是研究p进群或李型有限群的光滑表示的某些区别问题。考虑一个约化的p进群或有限的李型群G和G的一个闭子群H以及G的两个光滑（复）不可约表示\pi和H的\sigma。最基本的情况是\sigma是平凡的表示。我们感兴趣的问题是，当Hom_H（\pi，\sigma）非零时，它的维数是多少。在最有趣的情况下，这个维度等于0或1。这类问题由来已久。一个非常经典的例子是当H是G的极大紧开子群时，例如G=GL（n，Q_p）和H=GL（n，Z_p）。由这个H区分的表示被称为（H-）球面的，并且重数空间至多是一维的这一事实是从G的紧支撑函数的球面Hecke代数H（G，H）是阿贝尔的这一事实得出的，这些函数是H-双不变的。另一个著名的例子是Whittaker（或一般）表示的情况，在这种情况下，H是Borel子群的幂单根（当G是拟分裂的），\sigma是H的非退化特征标[CS 80]。一个特别有趣的情况是当\pi是G的岩堀不动点的光滑表示时。证明了这类表示的范畴等价于Iwahori-Hecke代数H（G，I）上紧支撑I-双不变函数的模范畴.因此，我们可以从Iwahori-Hecke代数的角度来研究这样的表示。这种方法带来的主要挑战是将我们正在研究的关于光滑表示的问题转移到岩堀-赫克代数设置中的等价问题。本研究方法的新奇在于从岩堀-赫克代数的角度强调了这种代数方法。在本项目中，我们可以达到多个目标。例如，人们可以尝试推广[CS 16]的结果，他们考虑了惠特克模型的表示。在[CS 16]中，G是p-adic域上的Chevalley群，H是G的Borel子群的幂单根，σ是H的非退化特征标. Iwahori Hecke代数的等价问题是确定有限Hecke代数的限制中包含Steinberg模的单模。另一个目标可以是考虑G = GL（2n，\Q_p），H = SP（2n，\Q_p）和\sigma平凡特征的例子，这是之前在[OS 07]中通过不同方法研究的一个案例。作为一个相关的玩具例子，我们必须考虑对称群的不可约表示，其对超八面体子群的限制包含平凡表示。更有趣的是，通过考虑F_q上有限Lie型群的类似限制问题，可以得到一些启示。最雄心勃勃的目标将是找到一个岩堀-赫克代数的共同框架，适用于一个巨大的类的区别问题，其中包括作为特殊情况下，上述例子。