Last update: 25 April 2013

Representations of affine and graded Hecke algebras associated to Weyl groups play an important role in the Langlands correspondence for the admissible representations of a reductive $p\text{-adic}$ group. We work in the general setting of a graded Hecke algebra associated to any real reflection group with arbitrary parameters. In this setting we provide a classification of all irreducible representations of graded Hecke algebras associated to dihedral groups. Dimensions of generalized weight spaces, Langlands parameters, and a Springer-type correspondence are included in the classification. We also give an explicit construction of all irreducible calibrated representations (those possessing a simultaneous eigenbasis for the commutative subalgebra) of a general graded Hecke algebra. While most of the techniques used have appeared previously in various contexts, we include a complete and streamlined exposition of all necessary results, including the Langlands classification of irreducible representations and the irreducibility criterion for principal series representations.

The affine Hecke algebra is tightly connected to the geometry and representation theory of a semisimple Lie group. In fact, the representation theory of affine Hecke algebras provides a large piece of the Langlands correspondence for the admissible representation theory of a reductive $p\text{-adic}$ group [Bor1976, KLu0862716]. The affine Hecke algebra is also present in the geometry of a semisimple group via the equivariant K-theory of the Steinberg variety. This connection plays an important role in the Springer correspondence and the Langlands classification. Recent conjectures of Lusztig tie the representation theory of the affine Hecke algebra to the modular representation theory of semisimple Lie algebras in positive characteristic. So there are many good reasons to study the representations of affine Hecke algebras.

With appropriate definitions, the graded Hecke algebra is the associated graded algebra of the affine Hecke algebra. Lusztig [Lus1995] has shown that the representation theory of graded Hecke algebras of Weyl groups is essentially equivalent to the representation theory of affine Hecke algebras. In the same way that the affine Hecke algebra is connected to equivariant K-theory [KLu0862716, CGi1433132] the graded Hecke algebra is connected to equivariant cohomology [Lus1995].

This paper is a study of the combinatorial representation theory of graded Hecke algebras associated to finite real reflection groups (including the noncrystallographic cases). The geometric representation theory of these algebras has been studied in [Lus1988, Lus1989, Lus1995] and fundamental results have appeared in [HOp1997, Opd1995]. However, a wealth of information can be obtained with purely combinatorial techniques. Here we develop the combinatorial theory from elementary principles. Most of the techniques we use are known in the affine Hecke algebra setting but they are spread over various parts of the literature, and in several cases the generalization to the graded Hecke algebras for the crystallographic case is nontrivial. We have collected these results, streamlined them, proved them in the general setting that includes noncrystallographic graded Hecke algebras, and made an effort to produce an up-to-date presentation. This paper includes

- the Langlands classification of irreducible representations,
- the theory of principal series representations (including the irreducibility criterion),
- the theory of intertwining operators,
- the classification of all irreducible representations for rank two algebras (including all dihedral cases ${I}_{2}\left(n\right)\text{),}$
- the classification of irreducible calibrated representations, and
- proofs of two conjectures from [Ram1998-2].

The Langlands classification for graded Hecke algebras is due to Evens [Eve1996]. We have shortened his proof but the shorter proof does not differ in any essential ideas. Our proof of the irreducibility criterion for principal series modules is a graded Hecke algebra analogue of the proof given by Kato [Kat1982-2] for affine Hecke algebras. Proofs of this criterion for graded Hecke algebras have appeared in [Che1991-2, Kri1999] but our proof is more constructive and gives detailed information about the spherical vectors in the principal series modules.

To our knowledge, the theory of intertwining operators originates from the study of affine Hecke algebra representations in Matsumoto [Mat1977]. In recent years this theory has played an important role in the theory of orthogonal polynomials, in particular, the study of Macdonald polynomials [Che1995, Opd1995, KSa1997]. In this paper we do not view these operators as intertwiners between principal series representations but rather as local operators on the weight spaces of any representation $\left(\tau \text{-operators}\right)\text{.}$ This generalized approach is increasingly common in the theory of Macdonald polynomials [Mac1423624]. Though we do not know of a reference for this theory in its application to representations of graded Hecke algebras, certainly all of these techniques are now standard in the orthogonal polynomial literature.

The full classification of all irreducible representations for rank two graded Hecke algebras is given in Section 3. We include detailed analysis of the structure (dimensions of generalized weight spaces) for these representations and their Langlands parameters. This analysis extends and completes the work on representations of rank two graded Hecke algebras included as part of [Kri1995, HOp1997]. In [Kri1995] only one-parameter algebras were included and the classification was only complete for $n$ odd; we now include the two-parameter case that arises when $n$ is even and treat nonregular central characters. In [HOp1997], general graded Hecke algebras were considered but the representations classified were spherical and tempered. An important consequence of our rank two construction is that it establishes a “Springer correspondence” for all dihedral groups. This correspondence is given in the final part of Section 3. As in [Ram2002], we express the hope that the irreducible representations in the rank two case will provide the foundation for a combinatorial construction of all irreducible representations.

In Section 4 we classify the irreducible calibrated representations (those with a simultaneous eigenbasis for a large commutative subalgebra) of graded Hecke algebras. These results are graded Hecke algebra analogues of the results in [Ram1998]. In addition to the classification, we give an elementary combinatorial construction of all irreducible calibrated representations of graded Hecke algebras. This construction is a generalization of the (seminormal) construction of the irreducible representations of the symmetric group given by Alfred Young [You1931, You1934]. In our construction the local regions and their chambers take the role that partitions and standard tableaux play in the symmetric group construction. Otherwise, the formulas used in the construction of the irreducible calibrated modules are exactly the same as those used by Young.

In Section 5, we give proofs of two conjectures from [Ram1998-2] which describe the combinatorial structure of the weights of graded Hecke algebra modules. One of these conjectures was proved by Losonczy [Los1999] and we present a slightly simplified version of his proof here. We then prove the other conjecture with a short reduction to the statement proved by Losonczy and exploit the reduction procedure to obtain new information about the combinatorial weight structure. The conjectures in [Ram1998-2] were only stated for the case when the reflection group $W$ is crystallographic and our proofs only hold for this case. We give examples that show analagous statements do not hold in the noncrystallographic case.

We would like to thank E. Opdam for his comments and suggestions on the manuscript. A. Ram thanks the Newton Institute for the Mathematical Sciences for their
hospitality and support (under EPSRC Grant GRK99015) during the special program on *Symmetric functions and Macdonald polynomials*.

This is an excerpt of a paper entitled *Representations of graded Hecke algebras*, written by Cathy Kriloff
(Department of Mathematics, Idaho State University, Pocatello, Idaho 83209-8085) and Arun Ram.

Research of the first author supported in part by an NSF-AWM Mentoring Travel Grant. Research of the second author supported in part by National Security Agency grant MDA904-01-1-0032 and EPSRC Grant GR K99015.