The aim of this project is to parameterize, construct, and describe the irreducible modules of the affine and double affine Hecke algebras. The investigator intends to extract combinatorial structure from the non-semisimple representations of these algebras, a key example of which is the crystal graph structure on the irreducible representations of the affine Hecke algebra of type A. The representation theory of Hecke algebras has applications in many areas of mathematics. Hecke algebras appear naturally in the representation theory of semisimple p-adic groups and are also a tool in the study of the modular representation theory of reductive groups over finite fields. They have intimate connections to quantum groups, statistical mechanics, and knot theory. Double affine Hecke algebras were defined by Cherednik and used by him to prove certain conjectures of Macdonald. They have connections to harmonic analysis of symmetric spaces and the classical theories of hypergeometric functions and q-hypergeometric functions.
This is a project in representation theory, which is the study of symmetry. Representation theory gives us the tools to solve problems about any system that exhibits symmetry, and so has wide applications in chemistry, physics, computer science, and even within other areas of mathematics. The investigator will study the most basic objects whose symmetries are encoded in a structure called the Hecke algebra. One of her primary methods is to "glue" these basic objects together in such a way that their global controlling structure is apparent---much in the same way that organizing the elements into the periodic table gives us information about the shared chemical properties of halogens (or other groups).
