Representations of Reductive Groups and Étale Hessenberg Varieties

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The Langlands program, a central theme in contemporary number theory, comprises several predictions that provide an approach to the analytic study of solution sets of Diophantine equations using techniques from linear algebra to exploit hidden symmetries in these solution sets. Geometric representation theory brings geometric tools to bear in studying problems in linear algebra. This project will apply methods from geometric representation theory to first reduce the computation of certain integrals that arise in the Langlands program to a discrete problem in combinatorial geometry, and then to study the solution of this latter problem. The results are expected to deepen and extend understanding in number theory and geometric representation theory, areas of mathematics with connections to many other subjects, including cyber security and physics.In work already completed, the investigator has succeeded in reducing the computation of certain orbital integrals to the problem of counting rational points of étale Hessenberg varieties over finite fields. This project will extend and refine the earlier computation, as well as study étale Hessenberg varieties in their own right, specifically as strata in affine Springer fibers, and place them in a broader context of relative Springer theory.