The Unreasonable Effectiveness of the Affine Hecke Algebra

University Of Georgia Research Foundation Inc, Athens GA

Investigators

Abstract

Understanding symmetry is a very profitable endeavor. For instance, if reflective symmetry is available, then computations can be cut in half: only half of the object is needed to understand it fully. Conversely, a lack of symmetry can also limit whether a material or object can possess a certain property. The overall theme of this project is to better understand certain groups of symmetries by studying how they transform a one, two, three, or higher dimensional space. This is well-understood when working with algebraic spaces based on the real numbers, but much more complicated when working in the modular case, that is, when real numbers are replaced by a finite number system that depends on a fixed prime number. Modular arithmetic is computationally easy and very useful; for instance, it forms the basis of modern algorithms in coding theory and cryptography. However, the problem of understanding and classifying how symmetries transform algebraic spaces in the modular case is more difficult than for real numbers and is not as well-understood. Broadly speaking, this project will study symmetries of algebraic spaces in the modular case. More precisely, much of the representation theory of a reductive, connected algebraic groups is encoded in affine Hecke algebras. Affine Hecke algebras have two geometric interpretations: equivariant constructible sheaves on the affine flag variety and equivariant coherent sheaves on the Steinberg variety. These two interpretations are known to be equivalent when working with characteristic zero sheaves. In the modular case, this is not yet known. However, if it does hold, one would expect certain structures on the coherent side that have yet to be defined. The PI will define exotic t-structures on derived categories of coherent sheaves for varieties similar to the Springer resolution and extend the powerful theory developed for intersection cohomology sheaves to the coherent setting. The PI will also define an Iwahori--Whittaker model for the Satake category. Finally, the PI will explicitly construct some central sheaves (those associated to tilting representations) in the mixed affine Hecke category. This project will strengthen the current categorical understanding of the affine Hecke algebra which, will aid its application to modular representation theory. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.