## Representations of rank two affine Hecke algebras

Last update: 31 March 2013

## Introduction

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 [Ram1998]. I have made special effort to describe how the classification here relates to the classifications by Langlands parameters (coming from $p\text{-adic}$ group theory) and by indexing triples (coming from a $p\text{-analogue}$ of the Springer correspondence). There are several reasons for doing the details of this classification:

1. The proof of the one of the main results of [Ram1998] depends on this classification of representations for rank two affine Hecke algebras. Specifically, in the proof of Proposition 4.4 of [Ram1998], one needs to know exactly which weights can occur in calibrated representations. The reason that this naturally depends on a rank two classification is outlined in (d) below.
2. The examples here illustrate (and clarify) results of [Ram1998], [KLu0862716], [CGi1433132], [BMo1989], [Eve1996], [Kri1999], [HOp1997,HOp1996]. Much of the power of the combinatorial methods which are now available is evident from the calculations in this paper, especially when one compares with the effort needed in other sources (for example [Xi1994], Chapt. 11).
3. The explicit information here can be very useful for obtaining results on representations of $p\text{-adic}$ groups (see, for example, [Lus1983-2]).
4. One hopes that eventually there will be a combinatorial construction of all irreducible representations of all affine Hecke algebras. I expect that such a construction will 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.

The first section of this paper is a review of definitions and basic results about affine Hecke algebras and their representations. A few additional lemmas are proved in order to aid the proofs and constructions in later sections. The remainder of the sections detail the classification and construction of the irreducible representations of affine Hecke algebras of types ${A}_{1},$ ${A}_{1}×{A}_{1},$ ${A}_{2},$ ${C}_{2}$ and ${G}_{2}\text{.}$ In each case I have indicated how the results here relate to the “Langlands classification”, the classification of Kazhdan and Lusztig [KLu0862716], and the results in [Ram1998].

## Acknowledgements

This paper is part of a series [Ram1998,Ram1998-2,Ram0401326] [RRa1998,RRa1998-2] on representations of affine Hecke algebras. During this work I have benefited from conversations with many people. To choose only a few, there were discussions with S. Fomin, F. Knop, L. Solomon, M. Vazirani and N. Wallach which played an important role in my progress. There were several times when I tapped into J. Stembridge’s fountain of useful knowledge about root systems. D.-N. Verma helped at a crucial juncture by suggesting that I look at the paper of Steinberg. G. Benkart was a very patient listener on many occasions. H. Barcelo, P. Deligne, T. Halverson, R. Macpherson and R. Simion all gave large amounts of time to let me tell them my story and every one of these sessions was helpful to me in solidifying my understanding.

I single out Jacqui Ramagge with special thanks for everything she has done to help with this project: from the most mundane typing and picture drawing to deep intense mathematical conversations which helped to sort out many pieces of this theory. Her immense contribution is evident in that some of the papers in this series on representations of affine Hecke algebras are joint papers.

A portion of this research was done during a semester long stay at Mathematical Sciences Research Institute where I was supported by a Postdoctoral Fellowship. I thank MSRI and National Science Foundation for support of my research.

## Notes and References

This is an excerpt of a preprint entitled Representations of rank two affine Hecke Algebras, written by Arun Ram, Department of Mathematics, Princeton University, August 5, 1989.

Research supported in part by National Science Foundation grant DMS-9622985, and a Postdoctoral Fellowship at Mathematical Sciences Research Institute.