Last update: 9 March 2013
We now turn to the rank two affine Hecke algebras. There are four different rank two root systems, each generated by two simple roots. The root systems are distinguished by the angle between the two simple roots - either or (See [Hum1994] for more details.)
The case where the two simple roots are perpendicular deserves special mention at this point. The root system is said to have type since it is the union of two root systems of type and Since all the elements of are perpendicular to all the elements of is reducible. Algebraically, the reducibility of is reflected in the fact that the Weyl group of R is isomorphic to the direct product of the Weyl groups of and
At the level of the affine Hecke algebra, the reducibility also has a strong algebraic meaning. Specifically, let and be the affine Hecke algebras built on the root data of and respectively. Then and are subalgebras of which centralize each other. In fact, each is the full centralizer of each other in Then a form of Schur-Weyl duality shows that all the irreducible modules are of the form where and are irreducible and modules, respectively. Hence the irreducible are in bijection with pairs of irreducible modules over and Since, then, the representation theory of the type affine Hecke algebra is completely determined by the representation theory of the type affine Hecke algebra, we will make no further comment about the type case.
This is an excerpt from Matt Davis' Ph.D Thesis entitled Representations of Rank Two Affine Hecke Algebras at Roots of Unity, University of Wisconsin, 2010.