Unitary representations of reductive Lie groups

最近，建立了一种算法来描述由复连通约化代数群的实点产生的李群的酉对偶。有趣的是，该算法与实群之外的其他数学思想建立了联系（例如，扭曲的Lusztig-Vogan多项式和对旗变体的混合Hodge模块的研究）。导致该算法的新理论发展提出了许多令人兴奋的新方向，无论是在将算法提供的答案概念化方面，还是在新环境中推动这些想法（例如，通过研究p进域上约化群表示的函数关系）。在这项赠款支助下进行的项目专门用于调查这些方向。特别是，PI计划研究该算法在预测为自同构的表示上的行为。他还计划将算法的某些方面应用于梯度仿射Hecke的设置，从而获得关于p进群的酉表示的信息。李群（以挪威数学家Sophus Lie的名字命名）是20世纪许多数学发展的中心。它们是物理学家对对称性进行编码的语言，因此为了将它们应用于表现对称性的物理问题而受到了密切的研究。一些特殊的例子导致了信号处理方面的重大进展。在一个完全不同的方向上，罗伯特·朗兰兹使用李群作为语言来解决数论中的问题，其中的一些方面导致了现代密码学的应用。在该基金支持下进行的项目一方面限制了李群如何作为物理问题的对称性出现，另一方面限制了它们如何作为数论问题的对称性出现。因此，这些限制排除了这两种上下文中的候选解决方案，从而提供了对实际解决方案和上面提到的重要应用程序的强大洞察力。

Recently, an algorithm has been established to describe the unitary dual of a Lie group arising as the real points of a complex connected reductive algebraic group. Interestingly, the algorithm forges connections with other mathematical ideas outside of real groups (for example, twisted Lusztig-Vogan polynomials and the study of mixed Hodge modules on flag varieties). The new theoretical developments that led to the algorithm suggest many exciting new directions, both in conceptualizing the answer that the algorithm provides, as well as pushing the ideas in new settings (for example by studying functorial relationships with representations of reductive groups over p-adic fields). The projects undertaken under the support of this grant are devoted to investigating those directions. In particular, the PI plans to investigate the behavior of the algorithm on representations predicted to be automorphic. He also plans to adapt aspects of the algorithm to the setting of graded affine Hecke, and therefore obtain information about unitary representations of p-adic groups.Lie groups (named after the Norwegian mathematician Sophus Lie) are at the center of much of the mathematics developed in the twentieth century. They are the language in which physicists encode symmetry, and thus have been studied intensely for their applications to physical problems exhibiting symmetry. Particular examples have led to significant advances in signal processing. In a completely different direction, Robert Langlands used Lie groups as the language to attack problems in number theory, aspects of which have led to applications in modern cryptography. The projects undertaken under the support of this grant place limitations on how Lie groups can appear as symmetries of physical problems, on one hand, and limitations on how they can appear as symmetries of number theory problems, on the other. These limitation, therefore, rule out candidate solutions in both contexts, and thus provide powerful insight into the actual solutions and the important applications mentioned above.