Local Langlands Theory for Real Lie Groups

最自然的理解是在局部朗兰兹猜想的表示理论部分的几何框架中，这个框架起源于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对偶于相关对偶空间上的等变束的几何范畴。在这个公式中，亚瑟猜想可以被解释为暗示了自同构形式的局部分量与对偶空间的微局部几何之间的非常精确的关系（编码在某些等变反常束的特征变化中）。当局部场是阿基米德时，局部朗兰兹猜想是一个定理，但阿瑟猜想是无效的，因为它们远远不能提供自同态谱的猜想枚举。我们提出了一个方案，将局部朗兰兹猜想的框架推广到实数群，这些实数群是作为实数群的实数点的非代数“元”双重覆盖出现的。我们的方法基于代数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枝的局部志村对应。这使得我们可以在非代数的背景下继续研究亚瑟的一些思想。最后，在一个单独的项目中，我们提出了一系列重要的和计算能力强大的限制形式的特征变化计算可以采取。（这些计算与Arthur理论有关，适用于代数和非代数群。）我们所追求的限制让我们深入了解如何使亚瑟王猜想对真实群体有效。提出的研究涉及对半简单（实）李群理论的理解。这些对象以有限维结构的对称性最为自然地出现；修饰语“半简单”大致意味着李群不能被分解成更小的李群。一个具有历史意义的例子是三维旋转的李群（它记录了球对称氢原子的对称性）。人们经常对李群如何作用于无限维线性空间（如氢原子薛定谔方程的解空间）感兴趣。这样的动作称为表示。在20世纪60年代，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的惊人关系。这是值得注意的，因为它连接了两个主题-表征理论和数论-表面上是不相关的。随后，朗兰兹提出的算术参数（以及更多朗兰兹的思想）被理解为奇异代数变异的深刻几何理论的阴影。提议者的研究试图扩展朗兰兹的一些想法，以及随后由于亚瑟的改进，以扩大“非代数”李群的类别，而不是最初在“代数”环境中考虑的。（这里的区别有点微妙，无法精确描述，但足以说明，从许多角度来看，非代数设置已被证明是非常有趣的。）我们的研究分为表示理论部分和几何部分。在定性上，表示理论部分是基于非代数理论和代数理论之间的精确关系（揭示关于每一个理论的新信息），而几何部分则是基于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，聪明的本科生可以访问；我计划在这个组合理论的基础上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.