| The affine Hecke algebra H˜ is an algebraic object related to the group algebra of the Weyl group of a Lie group G, which is built, from the root data of G. We classify and construct all the simple representations of the affine Hecke algebra H˜ for all possible values of the parameter q, when G, and thus H˜, have rank two. In particular, we describe these representations when q is a root of unity. We first define H˜ and establish the basic combinatorial tools that will be used to analyze the representations of the algebra. After proving some preliminary general results, we give, for each rank two algebra, an analysis of the composition factors of principal series modules, which includes all simple H˜-modules.;Once the classification is completed, we discuss a geometric approach to the same classification which is not, well-understood in the root of unity case. We discuss how these classifications match up and describe the geometric classification explicitly using our own classification of the simple H˜-modules. |
