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.
