This award supports research on the representation theory of affine Hecke algebras and quantum groups. The principal investigator will work on completing the classification of the unramified representations of semisimple split p-adic groups, via graded Hecke algebras and equivariant homology. He will also continue the study of the parameter space of these representations from the point of view of intersection cohomology. Further, he will study canonical bases in enveloping algebras and the total positivity in reductive groups over the reals. Many different algebraic objects can be represented as algebraic sets of transformations of other algebraic objects. Through these representations their structure can be determined. This project is concerned with the representation theory of certain algebras. The study of these algebras has applications throughout mathematics and mathematical physics.
