Unitary representations of reductive Lie groups

最近，已经建立了一个算法来描述作为复连通约化代数群的真实的点产生的李群的酉对偶。有趣的是，该算法与真实的群之外的其他数学思想建立了联系（例如，扭曲的Lusztig-Vogan多项式和对旗簇上的混合霍奇模的研究）。导致该算法的新理论发展提出了许多令人兴奋的新方向，无论是在概念化算法提供的答案，还是在新的设置中推动这些想法（例如通过研究p-adic域上约化群表示的函关系）。 在该赠款支持下开展的项目致力于调查这些方向。 特别是，PI计划研究算法在预测为自守的表示上的行为。 李群（以挪威数学家索菲斯·李群命名）是世纪数学发展的中心，它是一个数学领域的分支，也是一个数学领域的分支。李群是由挪威数学家索菲斯·李群（英语：Sophus Lie）所组成的。 它们是物理学家编码对称性的语言，因此，它们在展示对称性的物理问题中的应用受到了广泛的研究。 特定的例子已经导致信号处理的显著进步。 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。 在这项资助下进行的项目限制了李群如何作为物理问题的对称性出现，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情况下的候选解决方案，从而为实际解决方案和上述重要应用提供了有力的见解。

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.