  This paper classifies and constructs explicitly all the irreducible
representations of affine Hecke algebras of rank two root systems. The methods
used to obtain this classification are primarily combinatorial and are, for the
most part, an application of the methods used in a recent preprint of Ram. A
special effort has been made special effort to describe how the classification
here relates to the classifications by Langlands parameters (coming from p-adic
group theory) and by indexing triples (coming from a q-analogue of the Springer
correspondence). The examples here illustrate and clarify results of Ram,
Kazhdan-Lusztig, Chriss-Ginzburg, Barbasch-Moy, Evens, Kriloff, and
Heckman-Opdam. Much of the power of the combinatorial methods which are now
available is evident from the calculations in this paper. One hopes that
eventually there will be a combinatorial construction of all irreducible
representations of all affine Hecke algebras. Such a construction may depend
heavily on the rank two cases. This idea is analogous to the way that the rank
two cases are the basic building blocks in the presentations of Coxeter groups
by ``braid'' relations and the presentations of Kac-Moody Lie algebras (and
quantum groups) by Serre relations.

Representation Theory