Distinction problems by Iwahori-Hecke algebras

该项目属于ESPRC代数研究领域。这个项目的一般主题领域是在当地朗兰兹方案的总体框架内的p进约化群的表示理论。基于对自同构形式的应用，本课题的目的是研究李型p进群或有限群的光滑表示的若干区别问题。考虑一个可约的p进或有限李型群G和G的闭子群H，以及G的光滑（复）不可约表示\pi和H的\sigma，最基本的情况是\sigma是平凡表示。我们感兴趣的问题是，当Hom＿H（\pi, \sigma）不为零时，它的维数是多少。在最有趣的情况下，这个维数等于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，如G=GL（n,Q＿p）和H=GL（n,Z＿p）。由这个H所区分的表示称为（H-）球面，复数空间最多是一维的事实是由G的紧支持函数（H-双不变）的球面赫克代数H（G，H）是阿贝尔的这一事实得出的。另一个著名的例子是Whittaker（或泛型）表示，在这种情况下，H是Borel子群的单能根（当G是拟分裂时），\sigma是H的一个非简并特征[CS80]。要考虑的一个特别有趣的情况是\pi是具有Iwahori固定向量的G的光滑表示。结果表明，这种表示的范畴等价于紧支持I-双不变函数的Iwahori-Hecke代数H（G，I）上的模的范畴。因此，我们可以从Iwahori-Hecke代数的角度来研究这种表示。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换为Iwahori-Hecke代数设置中的等效问题。这种研究方法的新颖之处在于从Iwahori-Hecke代数的角度强调了这种代数方法。在这个项目中，我们可以达到多个目标。例如，人们可以尝试推广[CS16]用惠特克模型考虑表征的结果。在[CS16]中，G是p进域上的Chevalley群，H是G的Borel子群的单幂根，\sigma是H的非简并特征。Iwahori Hecke代数设置中的等价问题是确定其对有限Hecke代数的约束包含Steinberg模的简单模。另一个目标可以是考虑G = GL(2n, \Q ＿p), H = SP（2n, \Q ＿p）和\sigma为平凡字符的例子，这是之前在[OS07]中通过不同方法研究过的一个案例。作为一个相关的小例子，我们必须考虑对称群的不可约表示，其对高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Lie型有限群的类似限制问题，我们可以得到一些启示。最雄心勃勃的目标是找到一个Iwahori-Hecke代数的通用框架，它适用于大量的区分问题，其中包括作为特殊情况的例子上面提到的例子。

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.