Local Langlands Theory for Real Lie Groups

这项研究最自然地被理解为局部朗兰兹猜想的表征理论部分的几何框架，该框架起源于Deligne和Kazhdan-Lusztig的工作。粗略地说，该猜想断言局部域上的约化代数群的表示理论是对偶于相伴对偶空间上的等变层的几何范畴的。在这个公式中，Arthur猜想可以被解释为提出了自同构形的局部分量和对偶空间的微局部几何之间的非常精确的关系(编码在某些等变倒叶的特征簇中)。当局域场是阿基米德时，局域朗兰兹猜想是一个定理，但亚瑟猜想在某种意义上是无效的，因为它们不能提供自同构谱的猜想计数。我们提出了一个方案，将局部朗兰兹猜想的框架推广到作为代数群实点的非代数“亚交”二重覆盖的实群。我们的方法是基于代数理论和非代数理论之间的复杂关系，这种关系可以解释为一种分支的局部Shimura对应。这使我们能够在非代数的环境中追寻亚瑟的一些思想。最后，在一个单独的项目中，我们提出了一系列对特征变化计算可以采用的形式的非平凡和计算上强大的限制。(对于代数群和非代数群，这些计算都与Arthur理论有关。)我们追求的限制给了我们使Arthur猜想对实群有效的洞察力。拟议的研究涉及理解半单(实)李群的理论。这些物体最自然地以有限维结构的对称性出现；修饰词“半简单”的意思是，粗略地说，李群不能被分成更小的李群。一个具有历史意义的例子是三维旋转的李群(它编码了球对称氢原子的对称性)。人们通常感兴趣的是李群如何作用于无限维线性空间(如氢原子的薛定谔方程的解空间)。这样的行为被称为表象。在20世纪60年代的S中，朗兰兹提出了半单李群的表示理论与一组本质上是算术起源的数据之间惊人的关系。这是值得注意的，因为它连接了两个表面上无关的学科--表象理论和数论。随后，朗兰兹提出的算术参数(以及更多的朗兰兹思想)被理解为奇异代数簇的深层几何理论的影子。这位提出者的研究试图将朗兰兹的一些思想，以及亚瑟的一些后续改进，推广到比最初的“代数”环境中所考虑的更大的一类“非代数”李群。(这里的区别有点太微妙了，不能说得太准确，但可以说，从许多角度来看，非代数设置已经被证明非常有趣。)我们的研究分为表象理论部分和几何部分。定性地，表示理论部分基于非代数理论和代数理论之间的精确关系(揭示关于每个理论的新信息)，而几何部分则开启上述参数集的潜在结构。值得一提的是，对于特定的例子，这些潜在的结构相当于一个可爱的(非常具体的)组合，对聪明的本科生来说是可以接触到的；我计划基于这个组合理论来指导本科生的研究项目。

The proposed research is most naturally understood in terms of the geometric framework of the representation theoretic part of the local Langlands conjecture, a framework which originated in the work of Deligne and Kazhdan-Lusztig. Roughly speaking, the conjecture asserts that the representation theory of a reductive algebraic group over a local field is dual to a geometric category of equivariant sheaves on an associated dual space. In this formulation, the Arthur conjectures may be interpreted as suggesting a very precise relationship between local components of automorphic forms and the microlocal geometry of the dual space (codified in the characteristic variety of certain equivariant perverse sheaves). When the local field is archimedian, the local Langlands conjecture is a theorem, but the Arthur conjectures are ineffective in the sense that they fall well short of providing a conjectural enumeration of the automorphic spectrum. We propose a project to extend the framework of the local Langlands conjecture to real groups that arise as nonalgebraic "metaplectic" double covers of the real points of algebraic groups. Our approach is predicated on an intricate relationship between the algebraic and nonalgebraic theories which may be interpreted as a kind of ramified local Shimura correspondence. This allows us to pursue some of Arthur's ideas in the nonalgebraic setting. Finally in a separate project we propose a series of nontrivial and computationally powerful restrictions on the form that the characteristic variety computations can take. (These computations are relevant for the Arthur theory for both algebraic and nonalgebraic groups.) The restrictions we pursue give insight into making the Arthur conjectures effective for real groups.The proposed research deals with understanding the theory of semisimple (real) Lie groups. These objects arise most naturally as symmetries of finite-dimensional structures; the modifier "semisimple" means, roughly speaking, that the Lie group cannot be broken into smaller Lie groups. An example of historical import is the Lie group of rotations in three dimensions (which codifies the symmetry of the spherically symmetric hydrogen atom). One is often interested in how Lie groups can act on infinite-dimensional linear spaces (like the solution space of the Schroedinger equation for the hydrogen atom). Such actions are called representations. In the 1960's, Langlands suggested an astonishing relationship between the representation theory of semisimple Lie groups and a set of data of essentially arithmetic origin. This is remarkable because it connects two subjects-representation theory and number theory-which are ostensible unrelated. Subsequently the arithmetic parameters that Langlands suggested (and much more of Langlands ideas) became understood as a shadow of a deep geometric theory of singular algebraic varieties. The proposer's research seeks to extend some of the ideas of Langlands, and consequent refinements due to Arthur, to a larger class of "nonalgebraic" Lie groups than originally considered in the original "algebraic" setting. (The distinction is a little too delicate to make precise here, but suffice it to say that the nonalgebraic setting has been demonstrated to be very interesting from many perspectives.) Our research has a representation theoretic part and a geometric part. Qualitatively the representation theoretic part is predicated on a precise relationship between the nonalgebraic and algebraic theories (revealing new information about each) and the geometric part turns on a latent structure of the parameter set mentioned above. It is worth mentioning that for particular examples, these latent structures amount to a lovely (and very concrete) combinatoric which is accessible to bright undergraduates; I plan to direct undergraduate research projects based on this combinatorial theory.