Local Langlands Theory for Real Lie Groups

拟议的研究是最自然的理解方面的几何框架表示理论的一部分，当地朗兰兹猜想，一个框架，起源于工作的德利涅和Kazhdan-Lusztig。 粗略地说，该猜想断言，局部域上的约化代数群的表示理论与相关对偶空间上的等变层的几何范畴是对偶的。 在这种表述中，亚瑟的理论可以解释为，自守形式的局部分量与对偶空间的微局部几何（被编入某些等变反常层的特征多样性中）之间存在着非常精确的关系。 当局部域是阿基米德时，局部朗兰兹猜想是一个定理，但亚瑟猜想是无效的，因为它们远远不能提供自守谱的代数枚举。 我们提出了一个项目，以扩大当地的朗兰兹猜想的框架，以真实的群体，出现非代数的“元”双覆盖的真实的点的代数群。 我们的方法是基于代数和非代数理论之间的复杂关系，可以解释为一种分歧的当地志村对应。 这使我们能够追求一些亚瑟的想法在非代数设置。 最后，在一个单独的项目中，我们提出了一系列非平凡的和计算强大的限制的形式，特征品种的计算可以采取。 （这些计算与代数群和非代数群的亚瑟理论有关。） 我们所追求的限制使亚瑟理论对真实的群有效。所提出的研究涉及理解半单（真实的）李群的理论。 这些对象最自然地出现在有限维结构的对称性中;修饰语“半单”的意思是，粗略地说，李群不能分解成更小的李群。 历史上重要的一个例子是三维旋转的李群（它规范了球对称氢原子的对称性）。 人们通常对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）感兴趣。 这种行为称为表示。 在20世纪60年代，朗兰兹提出了半单李群的表示理论和一组基本上源于算术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示论和数论，这是表面上无关。 后来，朗兰兹提出的算术参数（以及朗兰兹更多的思想）被理解为奇异代数簇的深刻几何理论的影子。 提议者的研究试图将朗兰兹的一些想法和亚瑟的后续改进扩展到比最初在“代数”设置中考虑的更大类的“非代数”李群。 (The这里的区别有点太微妙了，无法精确地说明，但可以说，从许多角度来看，非代数的设置已经被证明是非常有趣的。）我们的研究包括表示论部分和几何部分。 定性的表示理论部分是基于非代数和代数理论之间的精确关系（揭示每个理论的新信息），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（和非常具体的）组合，这是访问明亮的本科生;我计划直接本科生的研究项目的基础上，这一组合理论。

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.