Affine Hecke algebras and their representations

This is a survey paper about affine Hecke algebras. We start from scratch and discuss some algebraic aspects of their representation theory, referring to the literature for proofs. We aim in particular at the classification of irreducible representations. Only at the end we establish a new result: a natural bijection between the set of irreducible representations of an affine Hecke algebra with real parameters $\geq 1$, and the set of irreducible representations of the affine Weyl group underlying the algebra. This can be regarded as a generalized Springer correspondence with affine Hecke algebras.


Introduction
Affine Hecke algebras typically arise in two ways. Firstly, they are deformations of the group algebra of a Coxeter system (W, S) of affine type. Namely, keep the braid relations of W , but replace every quadratic relation s 2 = 1 (s ∈ S) by (0.1) (s − q s )(s + 1) = 0, where q s is a parameter in some field (usually we take the field C). That gives rise to an associative algebra H (W, q). Secondly, affine Hecke algebras occur in the representation theory of reductive groups G over p-adic fields. They can be isomorphic to the algebra of G-endomorphisms of a suitable G-representation. The classical example is the convolution algebra of compactly supported functions on G that are bi-invariant with respect to an Iwahori subgroup. In this way the representation theory of affine Hecke algebras is related to that of reductive p-adic groups. The interpretation of (affine) Hecke algebras as deformations of group algebras links them with quantum groups, knot theory and noncommutative geometry. Further, their relation with reductive groups makes them highly relevant in the representation theory of such groups and in the local Langlands program.
Both views of affine Hecke algebras build upon simpler objects: the Hecke algebras of finite Weyl groups. On the one hand such a finite dimensional Hecke algebra is a deformation of a group algebra, again with relations (0.1). On the other hand it appears naturally in the representation theory of reductive groups over finite fields. However, there is a crucial difference between the finite and affine cases: most aspects of finite dimensional Hecke algebras are easily described in terms of a Coxeter group, but that is far from true for affine Hecke algebras.
Of course there exists an extensicve body of literature on affine Hecke algebras and their representations, see the references to this paper for a part of it. The theory is in a good state, and (in our opinion) most of the important questions that one can ask about affine Hecke algebras have been answered.
Unfortunately, the accessibility of this literature is rather limited. A first difficulty is that several slightly different algebras are involved, with several presentations, and sometimes their connection is not clear. Further, a plethora of techniques has been applied to affine Hecke algebras: algebraic, analytic, geometric or combinatoric to various degrees. Finally, to the best of our knowledge no textbook treats affine Hecke algebras any further than their presentations.
With this survey paper we intend to fill a part of that gap. Our aim is an introduction to affine Hecke algebras and their representations, of which the larger part is readable for mathematicians without previous experience with the subject. To ease the presentation, we will hardly prove anything, and we will provide many examples. As a consequence, our treatment is almost entirely algebraic -the deep analytic or geometric arguments behind some important results are largely suppressed.
Let us discuss the contents in a nutshell. In the first section we work out the various presentations of (affine) Hecke algebras over C, and we compare them. Section 2 consists of explicit examples: we look at the most frequent affine Hecke algebras and provide an overview of their irreducible representations.
The third section is the core of the paper, here we build up the abstract representation theory of an affine Hecke algebra H. This is done in the spirit of Harish-Chandra's analytic approach to representations of reductive groups: we put the emphasis on parabolic induction, the discrete series and the large commutative subalgebra of H. To make this work well, we need to assume that the parameters q s of H lie in R >0 . With these techniques one can divide the set of irreducible representations Irr(H) into L-packets, like in the local Langlands program. To achieve more, it pays off to reduce from affine Hecke algebras to a simpler kind of algebras called graded Hecke algebras. This is like reducing questions about a Lie group to its Lie algebra.
More advanced techniques to classify Irr(H) are discussed in Section 4. In principle this achieves a complete classification, but in practice some computations remain to be done in examples.
In Section 5 we report on algebro-geometric approaches to affine Hecke algebras and graded Hecke algebras, largely due to Lusztig. In many cases, these methods yield beautiful constructions and parametrizations of all irreducible representations. Although we treat the involved (co)homology theories as a black box, we do provide a comparison between these constructions and the setup inspired by Harish-Chandra.
The final section of the paper is quite different from the rest, here we do actually prove some new results. The topic is the relation between an affine Hecke algebra H with parameters q s ∈ R ≥1 and its version with parameters q s = 1. The latter algebra is of the form C[X W ], where X is a lattice and W is the (finite) Weyl group of a root system R in X. We note that X W contains the affine Weyl/Coxeter group ZR W . There is a natural way to regard any finite dimensional H-representation π as a representation of C [W ], we call that the W -type of π.
Theorem A. (see Theorem 6.12) Let H be an affine Hecke algebra with parameters in R ≥1 (and a mild condition when R has components of type F 4 ). There exists a natural bijection ζ H : Irr(H) −→ Irr( X W ) such that the restriction of ζ H (π) to W is always a constituent of the W -type of π.
It is interesting to restrict Theorem A to irreducible X W -representations on which X acts trivially (those can clearly be identified with irreducible W -representations). The inverse image of these under ζ H is the set Irr 0 (H) of irreducible tempered H-representations "with real central character", see Paragraph 6.1. When H is of geometric origin, Irr 0 (H) is naturally parametrized by data as in Lusztig's generalization of the Springer correspondence with intersection homology, see Paragraph 5.2. In that case the bijection ζ H : Irr 0 (H) → Irr (W ) becomes an instance of the generalized Springer correspondence. Moreover, for such geometric affine Hecke algebras the whole of Irr(H) admits a natural parametrization in terms of data that are variations on see Paragraph 5.3. Via this parametrization, ζ H becomes a generalized Springer correspondence for the (extended) affine Weyl group X W .
With that in mind we can regard Theorem A, for any eligible H, as a "generalization of the Springer correspondence with affine Hecke algebras". This applies both to the finite Weyl group W and the (extended) affine Weyl group X W .
Of course the selection of topics in any survey is to a considerable extent the taste of the author. To preserve a reasonable size, we felt forced to omit many interesting aspects of affine Hecke algebras: the Kazhdan-Lusztig basis [KaLu1], asymptotic Hecke algebras [Lus7], unitary representations [BaMo1,Ciu], the Schwartz and C *completions of affine Hecke algebras [Opd2,DeOp1], homological algebra [OpSo1,Sol5], formal degrees of representations [OpSo2,CKK], spectral transfer morphisms [Opd3,Opd4] and so on. We apologize for these and other omissions and refer the reader to the literature.

Definitions and first properties
1.1. Finite dimensional Hecke algebras.
Let (W, S) be a finite Coxeter system -so W is a finite group generated by a set S of elements of order 2. Moreover, W has a presentation Let q : S → C be the function s → q s . We define a new C-algebra H(W, q) which has a vector space basis {T w : w ∈ W }. Here T e is the unit element and there are quadratic relations (T s + 1)(T s − q s ) = 0 s ∈ S and braid relations (1.3) T s T s T s · · · m(s,s ) terms = T s T s T s · · · m(s,s ) terms .
Equivalent versions of these quadratic relations are (1.4) T 2 s = (q s − 1)T s + q s T e and T −1 s = q −1 s T s + (q −1 s − 1)T e (the latter only when q s = 0). For q = 1, the relations (1.3) become the defining relations of the Coxeter system (W, S), so H(W, 1) = C [W ]. We say that H (W, q) has equal parameters if q s = q s for all s, s ∈ S.
The condition (1.2) is necessary and sufficient for the existence of an associative unital algebra H (W, q) with these properties [Hum,]. It is known as a generic algebra or a Hecke algebra. Such algebras appear for instance in the representation theory of reductive groups over finite fields [Iwa1,HoLe1,HoLe2], in knot theory [Jon] and in combinatorics [HiTh, HST].
Further H (W, q) has the structure of a symmetric algebra: it carries a trace τ (T w ) = δ w,e , an involution T * w = T w −1 and a bilinear form (x, y) = τ (x * y). These also give rise to interesting properties [GePf], which however fall outside the scope of this survey.
Without the trace τ , the representation theory of finite dimensional Hecke algebras is quite easy. To explain this, we consider a more general situation.
Let G be any finite group. By Maschke's theorem the group algebra C[G] is semisimple. Let {T g : g ∈ G} be its canonical basis, and k = C[x 1 , . . . , x r ] a polynomial ring over C. Let A be a k-algebra whose underlying k-module is k [G] and whose multiplication is defined by (1.5) T g · T h = w∈G a g,h,w T w for certain a g,h,w ∈ k. For any point q ∈ C r we can endow the vector space C [G] with the structure of an associative algebra by (1.6) T g · q T h = w∈G a g,h,w (q)T w We denote the resulting algebra by H(G, q). It is isomorphic to the tensor product A ⊗ k C where C has the k-module structure obtained from evaluating at q. Assume moreover that there exists a q 0 ∈ C r such that We express the rigidity of finite dimensional semisimple algebras by the following special case of Tits' deformation theorem, see [Car, or [Iwa2,Appendix].
Theorem 1.1. There exists a polynomial P ∈ k such that the following are equivalent : • P (q) = 0, • H(G, q) is semisimple, In other words: when H(G, q) is semisimple it is isomorphic to C [G], and otherwise the algebra H (W, q) has nilpotent ideals and looks very different from C [G]. The simplest case where the latter occurs is already G = {e, s}. Namely, for q = −1 we get the Hecke algebra (1.7) H(G, −1) = C[T s + 1]/(T s + 1) 2 .
Let us discuss how it works out for an arbitrary finite Coxeter system (W, S) and a parameter function q as before. For an element w ∈ W with a reduced expression s 1 s 2 · · · s r we put (1.8) q(w) = q s 1 q s 2 · · · q sr . This is well-defined by the presentation of W and condition (1.2). We want to know under which conditions the algebra H(W, q) is semisimple. For such groups the polynomials P (q) of Theorem 1.1 have been determined explicitly. If we are in the equal label case q(s) = q ∀s ∈ S then we may take (1.9) P (q) = q w∈W q (w) except that we must omit the factor q if W is of type (A 1 ) n , see [GyUn]. More generally, suppose that if (W, S) is irreducible and S consists of two conjugacy classes, with parameters q 1 and q 2 . Gyoja [Gyo,p. 569] showed that in most of these cases we may take (1.10) P (q 1 , q 2 ) = q |W | 1 q 2 W (q 1 , q 2 )W (q −1 1 , q 2 ) W (q 1 , q 2 ) = w∈W q(w) So generically there is an isomorphism (1.11) H(W, q) ∼ = C [W ].
In particular, for almost all q the representation theory of H (W, q) is just that of W (over C). When q 1 q 2 = 0 and W (q 1 , q 2 ) = 0, the subgroup of C × generated by q 1 and q 2 contains a root of unity different from 1. Hence the non-semisimple algebras H (W, q) are those with (at least) one q s equal to 0 and some for which q involves nontrivial roots of unity. Although these algebras can have an interesting combinatorial structure, their behaviour is quite different from that of affine Hecke algebras.

Iwahori-Hecke algebras.
Let (W, S) be any Coxeter system. As in the previous paragraph, we can assign complex numbers q s to the elements of S. When (1.2) is fulfilled, we can construct an algebra H(W, q) exactly as before. For q = 1 this is a group algebra of W , and hence, for q = 1, H (W, q) can be regarded as some deformation of C [W ]. However, the situation is much more complicated than when W is finite. Tits' deformation theorem does not work here, and in general many different q's can lead to mutually non-isomorphic algebras [Yan].
To get some grip on the situation, we restrict our scope from general Coxeter groups to affine Weyl groups. By that we mean Coxeter systems (W aff , S aff ) such that every irreducible component S i of S aff generates a Coxeter group W i of affine type. The affineness condition is equivalent to: the Cartan matrix of (W i , S i ) is positive semidefinite but not positive definite [Hum,§2.5,§4.7,§6.5]. Thus irreducible affine Weyl groups are classified by the Dynkin diagramsÃ n ,B n ,C n ,D n ,Ẽ 6 ,Ẽ 7 ,Ẽ 8 ,F 4 ,G 2 .
In the simplest caseÃ 1 , W aff is an infinite dihedral group, freely generated by two elements of order 2. Definition 1.2. An Iwahori-Hecke algebra is an algebra of the form H (W, q), where (W, S) is any Coxeter system. We say that H (W, q) is of affine type if (W, S) is an affine Weyl group.
Iwahori-Hecke algebras of affine type where discovered first by Matsumoto and Iwahori [Mat1,IwMa,Iwa2], in the context of reductive p-adic groups. For instance, let G be a split, simply connected, semisimple group over Q p and let I be an Iwahori subgroup of G. Then the convolution algebra C c (I\G/I) is isomorphic to H (W aff , q), where (W aff , S aff ) is derived from G and q s = p for all s ∈ S. This is called the Iwahori-spherical Hecke algebra of G, because its modules classify G-representations with I-fixed vectors.
The basic structure of an affine Weyl group W aff is described in [Hum,§4.2] and [Bou,§VI.2]. The set of elements whose conjugacy class is finite forms a finite index normal subgroup of W aff , isomorphic to a lattice. That lattice is spanned by an integral root system R, and W aff is the semidirect product of ZR and the Weyl group of R. In particular H (W aff , q) contains H(W (R), q) as a subalgebra. However, the embedding of W (R) in W aff is in general not unique.

Affine Hecke algebras.
Affine Hecke algebras generalize Iwahori-Hecke algebras of affine type. Instead of affine Weyl groups, we allow more general groups which are semidirect products of lattices and finite Weyl groups. The best way to do the bookkeeping is with root data, for which our standard references are [Bou, Hum].
Consider a quadruple R = (X, R, Y, R ∨ ), where • X and Y are lattices of finite rank, with a perfect pairing ·, · : X × Y → Z, • R is a root system in X, • R ∨ ⊂ Y is the dual root system, and a bijection R → R ∨ , α → α ∨ with α, α ∨ = 2 is given, • for every α ∈ R, the reflection If all these conditions are met, we call R are root datum. It comes with a finite Weyl group W = W (R) and an infinite group W (R) = X W (R), the extended affine Weyl group of R. Often we will add a base of R to R, and speak of a based root datum.
We stress that we do not require R to span the vector space X ⊗ Z R. We say that R is semisimple if R does span X ⊗ Z R. For non-semisimple root data R and R ∨ may even be empty. For instance, root datum R = (Z n , ∅, Z n , ∅) has infinite group W (R) = Z n , but no reflections.
More examples of root data (actually all) come from reductive groups, see [Spr]. Suppose that G is a reductive algebraic group, T is a maximal torus in G and R(G, T ) is the associated root system. Denote the character lattice of T by X * (T ) and its cocharacter lattice by X * (T ). Then is a root datum. We note that R(G, T ) is semisimple if and only if G is semisimple.
The group W (R) = X W acts naturally on the vector space X ⊗ Z R, X by translations and W by linear extension of its action on X. The collection of hyperplanes H α,n = {x ∈ X ⊗ Z R : α ∨ (x) = n} α ∈ R, n ∈ Z is W (R)-stable and divides X ⊗ Z R in open subsets called alcoves. Let W aff be the subgroup of W (R) generated by the (affine) reflections in the hyperplanes H α,n . This is an affine Weyl group, and X ⊗ Z R with this hyperplane arrangement is its Coxeter complex.
To construct Hecke algebras from root data, we need to specify a set of Coxeter generators of W aff . In this setting they will be affine reflections. Let ∆ be a base of R. As is well-known, it yields a set of simple reflections S = {s α : α ∈ ∆}, and (W, S) is a finite Coxeter system. (For a root datum of the form R(G, T ) as in (1.12), the choice of ∆ is equivalent to the choice of a Borel subgroup B ⊂ G containing T .) The base ∆ determines a "fundamental alcove" A 0 in X ⊗ Z R, namely the unique alcove contained in the positive Weyl chamber (with respect to ∆), such that 0 ∈ A 0 . The reflections in those walls of A 0 that contain 0 constitute precisely S. The set S aff of (affine) reflections with respect to all walls of A 0 forms the required collection of Coxeter generators of W aff .
Example 1.4. A part of the hyperplane arrangement for an affine Weyl group of typeÃ 2 , with A = A 0 , ∆ = {α, β} and S aff = {s α , s β , s γ }. We can make S aff more explicit. Let R ∨ max be the set of maximal elements of R ∨ , with respect to the base ∆ ∨ . It contains one element for every irreducible component This is the reflection of X ⊗ Z R in the hyperplane H α,1 , a wall of A 0 . Then S aff = S ∪ {s α : α ∨ ∈ R ∨ max }. We denote the based root datum (X, R, Y, R ∨ , ∆) also by R. Thus the sets S, S aff and the subgroup W aff ⊂ W (R) are determined by R.
Example 1.5. A couple of important instances, coming from the reductive groups P GL 2 and GL n : This is the same W (R) as in (1.12), but with a different set of simple affine reflections.
Like Iwahori-Hecke algebras, affine Hecke algebras involve q-parameters. Fix q ∈ R >1 and let λ, λ * : R → C be functions such that Definition 1.6. The affine Hecke algebra H(R, λ, λ * , q) is the vector space C[X] ⊗ C H(W, q) with the multiplication rules: • C[X] and H(W, q) are embedded as subalgebras, • for α ∈ ∆ and x ∈ X: When α ∨ / ∈ 2Y , the cross relation simplifies to Notice that here the right hand side lies in C[X] because It follows from [Lus3,§3] that H(R, λ, λ * , q) is really an associative algebra with unit element θ 0 ⊗ T e , and that the multiplication map is bijective. We say that H(R, λ, λ * , q) has equal parameters if When λ = λ * = 1, we omit them from the notation and write simply H(R, q).
In that setting we will often allow q to be any element of C × . We note that for λ = λ * = 0 or q = 1 we recover the group algebra C[X W ]. In particular, the only affine Hecke algebra associated to (X, ∅, Y, ∅) is C[X]. Affine Hecke algebras appear foremostly in the representation theory of reductive p-adic groups, see [BuKu1,Mor,Roc,ABPS2,Sol8]. They also have strong ties to orthogonal polynomials [Kir, Mac], which often run via double affine Hecke algebras [Che2]. This has lead to a whole family of Hecke algebras, with adjectives like degenerate, cyclotomic, rational, graded and (double) affine. We will only discuss one further member of this family, in paragraph 1.5.
As already explained, Tits' deformation theorem does not apply to affine Hecke algebras. But there is a substitute for the semisimplicity part of Theorem 1.1. Recall that a finite dimensional algebra A is semisimple (i.e. a direct sum of simple algebras) if and only its Jacobson radical Jac(A) (the intersection of the kernels of all simple modules) is zero. In that sense Theorem 1.1 admits a partial generalization, see [Sol1,Lemma 3.4] and [Mat2,(3.4.5)]: Lemma 1.7. The Jacobson radical of an affine Hecke algebra H(R, λ, λ * , q) is zero.

Presentations of affine Hecke algebras.
We will make the relations between the algebras in paragraphs 1.2 and 1.3 explicit. We start with the Bernstein presentation of an Iwahori-Hecke algebra of affine type.
Let W aff be an affine Weyl group with Coxeter generators S aff , and write it as Consider an Iwahori-Hecke algebra H(W aff , q) with parameters q s ∈ C.
Here q(x) 1/2 is defined via (1.8) and q 1/2 Not every affine Hecke algebra is isomorphic to an Iwahori-Hecke algebra, for instance C[X] is not. To be precise, an isomorphism as in Theorem 1.8 exists if and only if the root datum R is that of an adjoint semisimple group. To compensate for this difference in scope, we take another look at the structure of W (R) for an arbitrary based root datum R.
We know from [Hum,§4.4] that the length function of (W aff , S aff ) satisfies (1.17) (w) = number of hyperplanes H α,n that separate w(A 0 ) from A 0 .
We extend to a function W (R) → Z ≥0 , by decreeing that (1.17) is valid for all w ∈ W (R). Then is a subgroup of W (R). The group Ω acts by conjugation on W aff and that action stabilizes S aff (the set of reflections with respect to the walls of A 0 ). Moreover, since W aff acts simply transitively on the set of alcoves in X ⊗ Z R: Example 1.9. Let R be of type GL n , as in (1.14). Then Ω is isomorphic to Z, generated by ω = e 1 (1 2 · · · n). The action of ω on S aff = {s 0 , s 1 , . . . , s n−1 } is ωs i ω −1 = s i+1 (where s n means s 0 ).
Let q : S aff → C × be the parameter function determined by q, λ, λ * . The conditions on λ and λ * (before Definition 1.6) ensure that q is Ω-invariant. Hence the formula ω(T w ) = T ωwω −1 defines an algebra automorphism of H (W aff , q). This gives a group action of Ω on H (W aff , q). Recall that the crossed product algebra H (W aff Now we can formulate the counterpart of Theorem 1.8. The algebra H(W aff , q) Ω, with the multiplication rules (1.18), is called the Iwahori-Matsumoto presentation of H(R, λ, λ * , q).
We conclude this paragraph with yet another, more geometric, presentation of affine Hecke algebras. Let T be the complex algebraic torus Hom Z (X, C × ). By duality Hom(T, C × ) = X, and the ring of regular functions O(T ) is the group algebra C[X]. The group W acts naturally on X, and that induces actions on C[X] and on T .
From the above presentation it is easy to find the centre of these algebras [Lus3]: An advantage of Definition 1.11 is that this presentation can also be used if T is just known as an algebraic variety, without a group structure. In that situation H(T, λ, λ * , q) can be studied without fixing a basepoint of T , it suffices to have the W -action and the elements θ −α ∈ O(T ) × for α ∈ ∆. This is particularly handy for affine Hecke algebras arising from Bernstein components and types for reductive p-adic groups. Even more flexibly, the above presentation applies when O(T ) is replaced by a reasonable algebra of differentiable functions on T , like rational functions, analytic functions or smooth functions.
We already observed that Tits' deformation theorem fails for affine Hecke algebras. Nevertheless, apart from Lemma 1.7 there is another analogue of Theorem 1.1, obtained by replacing the centre of an affine Hecke algebra by its quotient field.
Let C(X) be the quotient field of C[X], that is, the field of rational functions on the complex algebraic variety T . The action of W on T gives rise to the crossed product algebra C(X) W . The quotient field of Z H(R, λ, λ * , q) ∼ = C[X] W is C(X) W , which is also the centre of C(X) W . We construct the algebra (1.20) Here the multiplication comes from the description on the left, and it is the same algebra as obtained from Definition 1.11 by substituting C(X) for O(T ).
For α ∈ ∆ we define an element ı • sα of (1.20) by 1.5. Graded Hecke algebras. Graded (affine) Hecke algebras were discovered in [Dri,Lus3]. They are simplified versions of affine Hecke algebras, more or less in the same way that a Lie algebra is a simplification of a Lie group.
Let a be a finite dimensional Euclidean space and let W be a finite Coxeter group action isometrically on a (and hence also on a * ). Let R ⊂ a * be a reduced root system, stable under the action of W , such that the reflections s α with α ∈ R generate W . These conditions imply that W acts trivially on the orthogonal complement of RR in a * .
In contrast with the previous paragraphs, R does not have to be integral and W does not have to be crystallographic. Thus we are dealing with root systems as in [Hum,§1.2]: just reduced and W -stable, no further condition. In particular the upcoming construction applies equally well to Coxeter groups of type H 3 , H 4 or I Write t = a ⊗ R C and let S(t * ) = O(t) be the algebra of polynomial functions on t. We choose a W -invariant parameter function k : R → C and we let r be a formal variable. We also fix a base ∆ of R.
Definition 1.13. The graded Hecke algebra H(t, W, k, r) is the vector space • the cross relation for α ∈ ∆ and ξ ∈ S(t * ): The grading is given by deg(w) = 0 for w ∈ W and deg(x) = deg(r) = 2 for Notice that for k = 0 Definition 1.13 yields the crossed product algebra Let R ∨ ⊂ a be the coroot system of R, so with α, α ∨ = 2 for all α ∈ R. For x ∈ t * the following relation holds in H(t, W, k, r): In fact (1.21) can be substituted for the cross relation in Definition 1.13, that suffices to determine the algebra structure uniquely. With such a simple presentation, it is no surprise that the graded Hecke algebras H(t, W, k) have more diverse applications than affine Hecke algebras. They appear in the representation theory of reductive groups over local fields, both in the p-case [BaMo1,BaMo2,Lus5] and in the real case [CiTr1,CiTr2]. Further, these graded Hecke algebras can be realized with Dunkl operators [Che1,Opd1], which enables them to act on many interesting function spaces.
The above construction can be modified a little, and still produce the same algebra. Namely, fix α ∈ R and ∈ R >0 . For all w ∈ W we replace k(wα) by k(wα) and wα by wα-that again gives a root system in the sense of [Hum,§1.2]. This operation preserves the cross relation in Definition 1.13, so does not change the algebra. Further, we can allow R to be non-reduced, as long as we impose in addition that k( α) = k(α) whenever > 0 and α, α ∈ R -that still gives the same graded Hecke algebras.
Similarly, we can scale all parameters k(α) simultaneously. Namely, scalar multiplication with z ∈ C × defines a bijection m z : t * → t * , which clearly extends to an algebra automorphism of S(t * ). From Definition 1.13 we see that it extends even further, to an algebra isomorphism . Notice that for z = 0 the map m z is welldefined, but no longer bijective. It is the canonical surjection . Algebras like H(t, W, k, r) are degenerations of affine Hecke algebras (the version where q is a formal variable) and arise in the study of cuspidal local systems on unipotent orbits in complex reductive Lie algebras [Lus2,Lus4,Lus6,AMS2].
More often one encounters versions of H(t, W, k, r) with r specialized to a nonzero complex number. In view of (1.22) it hardly matters which specialization, so it suffices to look at r → 1. The resulting algebra H(t, W, k) has underlying vector space C [W ] ⊗ C S(t * ) and cross relations Like for affine Hecke algebras, we see from (1.23) that the centre of H(t, W, k) is As a vector space, H(t, W, k) is still graded by deg(w) = 0 for w ∈ W and deg(x) = 2 for x ∈ t * \ {0}. However, it is not a graded algebra any more, because (1.23) is not homogeneous in the case ξ = α. Instead, the above grading merely makes H(t, W, k) into a filtered algebra. The graded algebra obtained from this filtration is obtained by setting the right hand side of (1.23) equal to 0. In other words, the associated graded of H(t, W, k) is the crossed product algebra The algebras H(t, W, 0) and H(t, W, k) with k = 0 are usually not isomorphic. But there is an analogue of Tits' deformation theorem, similar to Proposition 1.12. Let Q(S(t * )) be the quotient field of S(t * ), that is, the field of rational functions on t. It admits a natural W -action, and the centre of the crossed product Q(S(t * )) W is Q((St * )) W . Using (1.24) we construct the algebra Q(S(t * )) W ⊗ S(t * ) W H(t, W, k), which as vector space equals In there we have elements Proposition 1.14. [Lus3, §5] (a) The map s α →ĩ sα extends to a group homomorphism is an algebra isomorphism. In particular Graded Hecke algebras can be decomposed like root systems and reductive Lie algebras. Let R 1 , . . . , R d be the irreducible components of R. W, k). Hence the representation theory of H(t, W, k) is more or less the product of the representation theories of the tensor factors in (1.25). The commutative algebra O(z) ∼ = S(z * ) is of course very simple, so the study of graded Hecke algebra can be reduced to the case where the root system R is irreducible.

Irreducible representations in special cases
The most elementary instance of an affine Hecke algebra is with empty root system R. The algebra associated to the root datum (X, ∅, Y, ∅) is just the group algebra C[X] of the lattice X. Its space of irreducible complex representations is Since C[X] is a subalgebra of H((X, R, Y, R ∨ ), λ, λ * , q) for any additional data R, λ, λ * and q, (2.1) is a good starting point for the representation theory of any affine Hecke algebra. We will discuss the affine Hecke algebras that appear most often, and we construct and classify all their irreducible representations.
Let R = (X, R, Y, R ∨ , ∆) be an arbitrary based root datum and take the parameters λ = λ * = 0. Then q s = 1 for all s ∈ S aff , and We denote the onedimensional representation of C[X] associated to t ∈ T by C t . Then ind More generally, consider any C[X] W -representation (π, V ) that is generated by the subspace Then V = w∈W π(w)V t and π(w Corollary 2.1. The functor ind Wt induces an equivalences between the following categories: • C[X] W t -representations on which C[X] acts via the character t, • C[X] W -representations V that are generated by V t . The first category in Corollary 2.1 is naturally equivalent with the category of W trepresentations. We conclude that, for every irreducible W t -representation (ρ, V ρ ), the C[X] W -representation π(t, ρ) := ind is irreducible. For comparison with later results we point out that π(t, ρ) is a direct summand of the induced representation and that Res .
The next result goes back to Frobenius and Clifford, see [RaRa, Appendix] for a modern account.
Theorem 2.2. Every irreducible C[X] W -representation is of the form π(t, ρ) for a t ∈ T and a ρ ∈ Irr(W t ). Two such representations π(t, ρ) and π(t , ρ ) are equivalent if and only if there exists a w ∈ W with t = w(t) and ρ = w · ρ.
Here w · ρ = ρ • Ad(w) −1 : wW t w −1 → Aut C (V ρ ). Theorem 2.2 involves a group action of W on the setT We call T //W :=T /W the extended quotient of T by W . Thus Theorem 2.2 gives a canonical bijection On Irr(C[X] W ) we have the Jacobson topology, whose closed sets are Via (2.2) we transfer this to a topology on T //W . Then the natural maps 2.2. Graded Hecke algebras with k = 0. As in Paragraph 1.5, we let W be a finite Coxeter group acting isometrically on a finite dimensional Euclidean space a. When k = 0, we do not need a root system R ⊂ a * to construct the algebra Considerations with Clifford theory, exactly as in the previous paragraph, lead to: Two such representations π(ν, ρ) and π(ν , ρ ) are equivalent if and only if there exists a w ∈ W with ν = w(ν) and ρ = w · ρ.

Iwahori-Hecke algebras of type A 1 .
Consider the based root datum It has an affine Weyl group W aff = W (R) of type A 1 , with Coxeter generators S aff = {s α , s α : x → 1 − x}. Affine Hecke algebras of the H(R, λ, λ * , q), for various parameters λ(α) and λ * (α), appear often in the representation theory of classical p-adic groups [GoRo, MiSt]. We will work out the irreducible representations of H(R, λ, λ * , q) in detail. To avoid singular cases, we assume throughout this paragraph that Recall that the case λ(α) = λ * (α) = 0 was already discussed in Paragraph 2.1. We abbreviate It is not difficult to see [Mat1] that every irreducible H-representation is a quotient of ind H C[X] (C t ) for some t ∈ T . By (1.16) this induced representation has dimension two. Using the Iwahori-Matsumoto presentation H(W aff , q), two onedimensional representations can be written down immediately. Firstly the trivial representation, given by triv(T sα ) = q λ(α) = q 1 , triv(T s α ) = q λ * (α) = q 0 , and secondly the Steinberg representation, defined by When H is the Iwahori-spherical Hecke algebra of SL 2 over a p-adic field F , these two representations correspond to the trivial and the Steinberg representations of SL 2 (F ) -as their names already suggested.
Via evaluation at 1 ∈ Z, we identify T = Hom Z (X, C × ) with C × . By Theorem 1.8 θ 1 = q −1/2 0 q −1/2 1 T s α T sα . For the trivial and Steinberg representations that means triv(θ 1 ) = q Therefore, as C[X]-representations: There are no further relations between the irreducible representations ind H C[X] (C t ). (c) The algebra H has precisely four other irreducible representations: triv, St and two that we call π(−1, triv), π(−1, St). They have dimension one and fit in short Remark. By the conditions (2.3), the four special values of t are all different, except that the last two coincide if q 1 = q 0 .
Then Cp + and Cp − are the only nontrivial H(W, q)-submodules of ind H C[X] (C t ). It follows that, whenever ind H C[X] (C t ) is irreducible as H-module, Cp + or Cp − is an H-submodule. We test for which t ∈ T this happens. The cross relation in H gives In ind H C[X] (C t ) we get (2.5) This can only be a scalar multiple of p + if tq Similarly we compute in H: In ind H C[X] (C t ) that leads to (2.6) If this is a scalar multiple of p − , then t −1 q We conclude that for On the other hand, when t equals one of these special values, the above calculations in combination with the fact that θ 1 generates C[X] imply that ind H C[X] (C t ) does have a onedimensional H-submodule. (b) Consider the element . It lies in the version of H = H(T, λ, λ * , q) obtained from Definition 1.13 by replacing O(T ) with C(X). By direct calculation in that algebra: By Frobenius reciprocity, for such t: In particular this shows that As T sα and θ 1 generate H, it follows that here Cp − is the Steinberg representation.
Again, since θ 1 and T sα generate H, we can conclude that ind H C[X] (C q 1/2 0 q 1/2 1 )/Cp − is the trivial representation of H.

The calculations around (2.6) show that ind
By (2.8) it has the same restriction to H(W, q) as the Steinberg representation, which explains our notation π(−1, St) for this Cp − . The quotient As H(W, q)-representation it is isomorphic to Cp + ∼ = triv, and therefore we write

Analogous considerations apply to ind
).
An important special case is q 1 = q 0 = q. When F is a non-archimedean local field with with residue field of order q, H(W aff , q) = H(R, q) arises as the Iwahorispherical Hecke algebra of SL 2 (F ).
The algebra H(R, q) is the simplest example of a true affine Hecke algebra, not isomorphic to some more elementary kind of algebra. For this algebra Theorem 2.4 says: , with π(−1, triv) and π(−1, St) irreducible and inequivalent.
• There are only two other irreducible H(R, q)-representations, triv and St, which both occur as subquotients of ind . This classification works for almost all q ∈ C × , only 1 and −1 are exceptional. (In view of (1.7), that is hardly surprising.) For H(R, −1) the trivial representation coincides with π(−1, triv) and the Steinberg representation coincides with π(−1, St). Consequently H(R, −1) has only two onedimensional representations. Apart from that, the above statements are valid when q = −1.
Although the definition and formulas for H(R, q) look considerably simpler than those for H(R, λ, λ * , q), in the end the latter is hardly more difficult. In terms of the induced representations ind H C[X] (C t ), the only differences occur when t ∈ −1, −q The root datum of type GL n is Via t → (t(e 1 ), . . . , t(e n )) we identify T with (C × ) n . We saw in Section 1 that All the simple affine reflections from S aff are Ω-conjugate, so q s = q s for s, s ∈ S aff . Call this parameter q and consider the affine Hecke algebra H n (q) := H(R n , q) of type GL n . The primary importance of such algebras is that they describe every Bernstein block in the representation theory of GL n (F ) for a non-archimedean local field F [BuKu1]. In all those cases q is a power of a prime number, but just as affine Hecke algebra that is not necessary. We do not even have to require that q ∈ R >1 , the representation theory of H n (q) looks the same for every q ∈ C × which is not a root of unity. The irreducible representations of GL n (F ) were classified (in terms of supercuspidal representations) by Zelevinsky [Zel, BeZe]. That classification follows a combinatorial pattern involving certain "segments, and the irreducible representations of H n (q) exhibit the same pattern. We formulate their classification in terms intrinsic to Hecke algebras. The algebra H 1 (q) = C[Z] has already been discussed, so we may assume that n ≥ 2 (which ensures that S aff is nonempty).
Like in Paragraph 2.3 we start with the trivial and the Steinberg representations. By definition This does not yet determine the values of these representations on T ω , to fix that one requires in addition The terminology is motivated by Iwahori-spherical representation of GL n over a p-adic field F : triv and St correspond to the epynomous representations of GL n (F ).
In contrast with the Iwahori-Hecke algebra of type A 1 , the trivial and Steinberg representations of H n (q) come in a families of representations, parametrized by C × . This is implemented as follows. For z ∈ C × we put t z = (z, z, . . . , z) ∈ T Sn . We define the H n (q)-representation triv ⊗ t z by (2.11) and This is possible because α(t z ) = 1 for all α ∈ R, so that t z ∈ Hom(X, C × ) is trivial on X ∩ W aff . Similarly we define the onedimensional H n (q)-representation St ⊗ t z , by requiring (2.11) and For H 2 (q) the above onedimensional representations, in combination with the induced representations ind , already exhaust Irr(H 2 (q)). With arguments very similar to those in Paragraph 2.3 one can show: This representation is isomorphic to ind , but apart from that there are no relations between the irreducible representations of the form ind The only other irreducible H 2 (q)-representations are triv ⊗ t z and St ⊗ t z with z ∈ C × . They are mutually inequivalent and fit in short exact sequences The irreducible representations from Theorem 2.5 come in three kinds, but there is a natural way to gather them in two families: • The twists St ⊗ t z of the Steinberg representation, parametrized by t z ∈ T S 2 .
• A family parametrized by T /S 2 , which for almost all W t ∈ T /W has the member ind When that representation happens to be reducible, we take the unique representative t of W t with |α(t)| ≥ 1 and we assign to W t the unique irreducible quotient of ind The irreducible representations of H n (q) with n ≥ 2 can be parametrized with a longer list of similar families. We sketch the construction and classification, as translated from [Zel], stepwise.
• Choose a partition n = (n 1 , n 2 , . . . , n d ) of n, where n i ≥ 1 (but the sequence (n i ) need not be monotone).
• For such z it follows from the Langlands classification that π( n, z) has a unique irreducible quotient. That is the irreducible H n (q)-representation associated with ( n, z). • The irreducible H n (q)-representations assigned to ( n, z) and ( n , z ) are equivalent if and only if ( n, z) and ( n , z ) are S n -associate.
In the above parametrization z i ∈ (C × ) n i Sn i , so z can be regarded as a diagonal matrix in GL n (C). The partition n determines a Levi subgroup and z ∈ Z(M ). Let u m be a regular unipotent element of GL m (C) (it is unique up to conjugation). Then u := (u n 1 , u n 2 , . . . , u n d ) is a regular unipotent element of M , and z u is the Jordan decomposition of an element of GL n (C). Up to conjugacy, every element of GL n (C) has this shape, for some z (unique up to S n -association) and u. This setup leads to: Theorem 2.6. There exists a canonical bijection between the following sets: • conjugacy classes in GL n (C),

Parabolic induction.
In the representation theory of reductive groups, a pivotal role is played by parabolic induction. An analogous operation exists for affine Hecke algebras, and it will be crucial in large parts of this paper.
Given a based root datum R = (X, R, Y, R ∨ , ∆) and a subset P ⊂ ∆, we can form the based root datum We record the special cases R ∆ = R and R ∅ = (X, ∅, Y, ∅, ∅).
Let W P be the Weyl group of R P . Any parameter functions λ, λ * for R restrict to parameter functions for R P , and is a subalgebra of H = H(R, λ, λ * , q). This corresponds to the notion of a parabolic subgroup P of a reductive group G, and simultaneously to the notion of a Levi factor of P-for affine Hecke algebras the unipotent radical of P is more or less automatically divided out. Notice that H and H P share the same commutative subalgebra C[X] = O(T ). By (1.16), as vector spaces Parabolic induction for representations of affine Hecke algebras is the functor ind H H P : Mod(H P ) → Mod(H), for any parabolic subalgebra H P of H. We also have a "parabolic restriction" functor, that is just restriction of H-modules to H.
The link with parabolic induction for reductive p-adic groups is made precise in [Sol7,§4.1]. In that setting parabolic restriction for Hecke algebras corresponds to the Jacquet restriction functor (but with respect to a parabolic subgroup opposite to the one used for induction).
In practice we often precompose parabolic induction with inflations of representations from a quotient algebra of H P . To define it, we write In terms of reductive groups, the semisimple root datum R P corresponds to the maximal semisimple quotient of a parabolic subgroup P of G. The parameter functions λ and λ * remain well-defined for R P , so there is an affine Hecke algebra

In particular we have
From Definition 1.6 we see that the quotient map Via this quotient map we will often (implicitly) inflate H P -representations to H Prepresentations. To incorporate representations of H P that are nontrivial on {θ x : x ∈ X ∩ (P ∨ ) ⊥ }, we need more flexibility. Write so that T P T P = T and T P ∩ T P is finite. Every t ∈ T P gives rise to an algebra automorphism This operation corresponds to twisting representations of a reductive group by an unramified character. For any H P -representation (π, V ) and any t ∈ T P , we can form the H P -representation and it has dimension [W : W P ] dim(V σ ). Let us discuss how parabolic induction works out for the subalgebra C[X] of H.
When V t = 0, we call t a C[X]-weight of π, and V t its generalized weight space. We denote the set of C[X]-weights of (π, V ) by Wt(π) or Wt(V ).
which explains why we call V t a weight space and not a generalized weight space. If V has finite dimension, then we can triangularize the commuting operators π(θ x ) simultaneously, and we find Hence, whenever π admits a central character cc(π), we have The set of C[X]-weights of a representation behaves well under parabolic induction. To describe the effect, let W P be the set of shortest length representatives for W/W P .
Since H has finite rank as a module over its centre C[X] W , every irreducible Hrepresentation has finite dimension. Hence π admits at least one C[X]-weight, say t. By Frobenius reciprocity (C t , π) = 0. We conclude that:

Most of the time ind
To make that precise, consider the following rational functions on T : There are a few ways in which c α can simplify: The parabolically induced representations can all be realized on the same vector space H (W, q). In fact, they are isomorphic to H(W, q) as H(W, q)-modules. In principle, the entire representation theory of H can be uncovered by analysing the family of representations (3.5). We already did that successfully for W aff of type A 1 in paragraph 2.3. However, this direct approach is very difficult in general. Indeed, while the irreducible representations of H have been classified in several ways, the finer structure of ind H C[X] (C t ) (e.g. a Jordan-Hölder sequence or the multiplicity with which irreducible representations appear) is not always known.

Tempered representations.
An admissible representation of a reductive group G over a local field is tempered if all its matrix coefficents have moderate growth on G, see [Wal,§III.2] and [Kna,§VII.11]. This notion has several uses: • the irreducible tempered G-representations form precisely the support of the Plancherel measure of G, • the Langlands classification of irreducible admissible G-representations in terms of irreducible tempered representations of Levi subgroups of G, • for general harmonic analysis on G, e.g. the Plancherel isomorphism. Analogous of all these well-known results have been established for affine Hecke algebras, see [Opd2,DeOp1]. In this paragraph we will discuss the second of the above three items.
Recall that every (finite dimensional) H-module (π, V ) has a set of C[X]-weights Wt(π) ⊂ T = Hom Z (X, C × ). To formulate the condition for temperedness in terms of weights, it will be convenient to abbreviate The complex torus T admits a polar decomposition Here the unitary part T un is the maximal compact subgroup of T and the positive part exp(a) is the identity component of the maximal real split subtorus of T . Notice that The acute positive cones in a are We define a * ,+ and a * ,++ similarly, so in particular X + = X ∩ a * ,+ . Next we we have the obtuse negative cones in a: Via the exponential map exp : t → T we get Definition 3.6. A finite dimensional H-representation (π, V ) is tempered if the following equivalent conditions are satisfied: Example 3.7.
• Consider the root datum R n of type GL n . Then The Steinberg representation has q (1−n)/2 , q (3−n)/2 , . . . , q (n−1)/2 as its only C[X]-weight, so it is tempered when q ≥ 1. On the other hand, the trivial In the Langlands classification we need a somewhat more general kind of Hrepresentation, for which we merely require that it becomes tempered upon restriction to H (W aff , q). This can also be formulated with a more relaxed condition on the weights. This involves the Lie subgroup T ∆ of T , whose Lie algebra is identified with t ∆ := R ⊥ ⊂ t.
Definition 3.9. A finite dimensional H-representation (π, V ) is essentially tempered if the following equivalent conditions are satisfied: When the root datum R is semisimple, essentially tempered is the same as tempered.

] and [Sol7, Lemma 2.3])
For any irreducible essentially tempered H-representation π, there exists t ∈ T ∆ such that π • ψ t is tempered and arises by inflation from a representation of H ∆ .
Example 3.11. Consider R n and H n (q) with q > 1. Then For z ∈ C × the H n (q)-representation St ⊗ t z from (2.13) has a unique C[X]-weight It is essentially tempered for all z ∈ C, and tempered if and only if |z| = 1.
In the Langlands classification we employ irreducible representations of H P , where P ⊂ ∆. We need some further notations: We say that (π, V ) ∈ Irr(H P ) is in positive position if cc(π) = tW P r with t ∈ T P,++ T un and r ∈ T P . By Lemma 3.10 this is equivalent to requiring that π = π •ψ t for some t ∈ T P,++ T P un and π ∈ Irr(H P ). Definition 3.12. A Langlands datum for H consists of a subset P ⊂ ∆ and an irreducible essentially tempered representation σ of H P in positive position.
Equivalently, it can be given by a triple (P, τ, t), where P ⊂ ∆, τ ∈ Irr(H P ) is tempered and t ∈ T P,++ T P,un . Then the associated H P -representation is σ = τ • ψ t .
The H-representations ind H H P (σ) and ind H H P (τ • ψ t ) are called standard. By Lemma 3.3 every standard H-module admits a central character. Now we can finally state the Langlands classification for affine Hecke algebras.
Some consequences can be drawn immediately: • L(P, σ) is tempered if and only if P = ∆ and σ ∈ Irr(H) is tempered (because L(∆, σ) = σ and Langlands data are unique). • In terms of Langlands data (P, τ, t) and (P , τ , t ), the irreducible H-representations L(P, τ, t) = L(P, τ • ψ t ) and L(P , τ , t ) = L(P , τ • ψ t ) are equivalent if and only if P = P and τ • t ∼ = τ • t . (The latter does not imply t = t , for t and t may still differ by an element of T P ∩ T P .) • For q = 1, Corollary 2.1 shows that every standard module is irreducible.
Hence the notions of irreducible representations and standard modules coin- Example 3.14. We work out the Langlands classification for H(R, q) with R of type A 1 . Its irreducible representations were already listed in paragraph 2.3. The irreducible tempered representations are: • the Steinberg representation, • the two representations π(−1, triv) and π(−1, St) which sum to ind H C[X] (C −1 ).
These representations exhaust the Langlands data (P, σ) with P = ∆ = {α}. For In this way we obtain every element of Irr(H(R, q))-as listed at the end of paragraph 2.3-exacly once, because T ++ T un is a fundamental domain for the action of W = s α on T ∼ = C × .
Theorem 3.13 provides a quick and beautiful way to classify Irr(H) in terms of the irreducible tempered representations of its parabolic subquotient algebras H P .
On the downside, it conceals the topological structure of Irr(H). For instance, with R of type A 1 as discussed above, there is a family of H-representations π(∅, triv, t) = ind H C[X] (C t ) with t ∈ T , irreducible for almost all t. The Langlands classification breaks it into two families, one with t ∈ T un and one with t ∈ T \ T un . In the next paragraph will see how this can be improved.
We end this paragraph with some useful extras about the Langlands classification. The central character of an irreducible H P -representation τ is an element cc(τ ) of T P /W P . Its absolute value |cc(π)|, with respect to the polar decomposition (3.6), lies in exp(a P )/W P , and log |cc(π)| ∈ a P /W P . We fix a W -invariant inner product on a. The norm of log |t| is the same for all representatives t ∈ T P of cc(τ ). That enables us to write cc(τ ) = log |t| for any t ∈ T P with W P t = cc(τ ).
The representation L(P, τ, t) appears with multiplicity one in π(P, τ, t) = ind H H P (τ • ψ t ). All other constituents L(P , τ , t ) of π(P, τ, t) are larger, in the sense that cc(τ ) > cc(τ ) . (c) Let W s ∈ T /W . Both {π ∈ Irr(H) : cc(π) = W s} and {π(P, τ, t) : (P, τ, t) Langlands datum with cc(τ )t ⊂ W s} are bases of the Grothendieck group of the category of finite dimensional Hmodule all whose O(T )-weights are in W s. With respect to a total ordering that extends the partial ordering defined by part (b), the transition matrix between these two bases is unipotent and upper triangular.

Discrete series representations.
In the representation theory of a reductive group G over a local field, Harish-Chandra showed that every irreducible tempered G-representation τ can be obtained from an irreducible square-integrable modulo centre representation δ of a Levi subgroup M of G, see [Kna,Theorem 8.5.3] and [Wal,Proposition III.4 More precisely, τ is a direct summand of the parabolic induction of δ. When the centre of M is compact, δ is an isolated point of the space of irreducible tempered M -representations, and it is called a discrete series representation. Then it is a subrepresentation of L 2 (M ).
Like for the Langlands classification, these results van be formulated and proven for affine Hecke algebras as well. For these purposes it is essential that q s ∈ R >0 for all s ∈ S aff . To achieve that, we require not only that q ∈ R >1 (as we already did), but also that This condition will be in force in the remainder of this paragraph. We note that the space a −− from (3.7) is empty unless R spans a * , and in that case and that gives rise to a + P , We say that π belongs to the discrete series if the following equivalent conditions hold: Further, we can π essentially discrete series if the following equivalent conditions hold: Example 3.17.
• The affine Hecke algebra H n (q) of type GL n has no discrete series, because its root datum R n is not semisimple. Here For any z ∈ C × , the twist St ⊗ t z = St • ψ tz of the Steinberg representation is essentially discrete series. In fact these are all irreducible essentially discrete series representations of H n (q) (recall that q > 1). • An affine Hecke algebra with λ = λ * = 0 does not have discrete series representations, apart from the case R = (0, ∅, 0, ∅), when the trivial representation of H = C is regarded as discrete series.
We condition (3.8) at hand, we can define a Hermitian inner product on H by declaring that {q(w) −1/2 T w } is an orthonormal basis. Let L 2 (W (R), q) be the Hilbert space completion of H with respect to this inner product-it is canonically isomorphic to L 2 (W (R)). By [Opd2,Lemma 2.22], every irreducible discrete series representation of H is isomorphic to a subrepresentation of the regular representation of H on L 2 (W (R), q).
In terms of representations of a reductive group G, "essentially discrete series" means that a representation has finite length, and that its restriction to the derived group of G is square-integrable. An essentially discrete series representation is tempered if and only if {θ x : x ∈ X ∩ (∆ ∨ ) ⊥ } acts on it by characters from T ∆ un . Theorem 3.18. [DeOp1, Theorem 3.22] and [Sol7, Lemma 1.3] (a) Let π be an irreducible tempered H-representation. There exist P ⊂ ∆ and a tempered essentially discrete series representation δ of H P such that π is a direct summand of ind H H P (δ ). (b) Let π be an irreducible essentially discrete series H P -representation. There exist t ∈ T P and a discrete series Clearly, a combination of Theorems 3.13 and 3.18 yields some description of Irr(H) in terms of parabolic induction and the discrete series of the subquotient algebras H P with P ⊂ ∆. We work this out in detail.
Definition 3.19. An induction datum for H is a triple ξ = (P, δ, t), where • P ⊂ ∆, • δ is an irreducible discrete series representation of H P , • t ∈ T P . We regard two triples ξ and ξ = (P , δ , t ) as isomorphic (notation ξ ∼ = ξ ) if P = P , t = t and δ ∼ = δ . Let Ξ be the space of such induction data, topologized by regarding P and δ as discrete variables and T P as a complex analytic variety. We say that ξ = (P, δ, t) is positive, written ξ ∈ Ξ + , when |t| ∈ T P + .
We already associated to such an induction datum the parabolically induced representation π(P, δ, t) = ind H H P (δ • ψ t ). By Proposition 3.8 (3.9) π(P, δ, t) is tempered ⇐⇒ t ∈ T P un . For ξ ∈ Ξ + we write P (ξ) = {α ∈ ∆ : |t(α)| = 1}. Then P ⊂ P (ξ). This set of simple roots is useful because it allows to break the process of parabolic induction in two steps: the first dealing only with essentially tempered representations and the second similar to the Langlands classification. More concretely, by Proposition 3.8 ind is essentially tempered.
For a given π ∈ Irr(H), there are in general several induction data ξ ∈ Ξ + such that π is a quotient of π(ξ). So, in contrast with the Langlands classificaiton, Proposition 3.20 does not provide an actual parametrization of Irr(H). To bring that goal closer, one has to analyse the relations between the various representations π(ξ) with ξ ∈ Ξ.
For u in the finite group we have an automorphism ψ u of H P and a similar automorphism ψ P,u of H P , given by Then (δ • ψ −1 P,u ) • ψ ut = δ • ψ t , so (3.10) π(P, δ • ψ −1 P,u , ut) = π(P, δ, t). Suppose that w ∈ W and w(P ) = P ⊂ ∆. Then is an algebra isomorphism, and it descends to an algebra isomorphism ψ w : H P → H P . Moreover, ψ w can be implemented as conjugation by the element (3.12) ı • w ∈ C(X) W ⊗ C[X] W H from Proposition 1.12, see [Sol4,(3.124)]. There is a bijection (3.13) By [Opd2,Theorem 4.33] it is isomorphism between that H-representations . For t in a Zariski-open dense subset of T P , I w specializes to an H-isomorphism (3.14) π(P, δ, t) ∼ − → π(w(P ), δ • ψ −1 w , w(t)). As such, I w : H ⊗ H P V δ → H ⊗ H P V δ is a rational map in the variable t ∈ T P (possibly with poles for some t ∈ T P ).
Proof. The above implies that, for t in a Zariski-open subset of T P , these two Hrepresentations have the same character. The vector space H ⊗ H P V δ does not depend on t and the character of π(P, δ, t) depends algebraically on t ∈ T P , so in fact π(P, δ, t) and π(w(P ), δ • ψ −1 w , w(t)) have the same character for all t ∈ T P . Since H is of finite rank as module over its centre, the Frobenius-Schur theorem [CuRe,Theorem 27.8] applies, and says that the characters of inequivalent irreducible H-representation are linearly independent functionals on H. As the character of π(P, δ, t) determines that representation up to semisimplification, it carries enough information to determines the multiplicities with which the irreducible representations appear in π(P, δ, t).
The next results are much deeper, for their proofs involve a study of topological completions of affine Hecke algebras [Opd2,DeOp1]. For a systematic bookkeeping of the H-isomorphisms (3.10) and (3.14), we put them in a groupoid W Ξ . Its base space is the power set of ∆, the collection of morphisms from P to P is and the composition comes from the group T W . This groupoid acts on the space of induction data Ξ as . Theorem 3.22. Let ξ = (P, δ, t), ξ = (P , δ , t ) ∈ Ξ + . The H-representations π(ξ) and π(ξ ) have a common irreducible quotient if and only if there exists a (w, u) ∈ W Ξ with (w, u)ξ = ξ .
Finally, everything is in place to formulate an extension of the Langlands classification that incorporates results about discrete series representations. The outcome is similar to L-packets in the local Langlands correspondence. Recall that (3.8) is in force.
Further, π is tempered if and only if t ∈ T un .
Conversely, however, π(P, δ, t) can have more than one irreducible quotient. So, like an L-packet for a reductive group can have more than one element, (P, δ, t) might be associated (in Theorem 3.24) to several irreducible H-representations.
• Consider R of type A 1 , H = H(R, q). Then W Ξ,∅∅ = {1, s α }, W Ξ,∆∆ = {1} and Theorem 3.24 works out as follows, where we take t or t −1 depending on whether |t| ≥ 1 or |t| ≤ 1: • Keep R of type A 1 , but consider H = H(R, 1) = C [W aff ]. In this case H does not have any discrete series representations, and the effect of Theorem 3.24 is: • For R of type GL n , H = H n (q), and P ⊂ ∆ given by a partition n = (n 1 , n 2 , . . . , n d ) of n, we have The only irreducible discrete series representations of H(R n,P , q) are St ⊗ t z = St • ψ P,tz with t z ∈ T P ∩ T P . Recall that An induction datum (P, St ⊗ t z , z) with z = (z 1 , . . . , z d ) ∈ T P is positive if and only if |z 1 | ≥ |z 2 | · · · ≥ |z n |, a condition which is preserved by the action of T P ∩ T P on such induction data. Hence, up W Ξ -association, it suffices to consider only positive induction data (P, St, t)-so with t z = 1. It is easily checked that every pair ( n, z) as in Theorem 2.6.b is S nassociate to a pair ( n, z) which is positive in the above sense, and that the latter is unique up to W Ξ -association. The irreducible H n (q)-representation attached to ( n, z) in paragraph 2.4 is a quotient of π(P, St, z) = π( n, z), just as in Theorem 3.24. Thus, for H n (q) Theorem 3.24 recovers Theorem 2.6: both parametrize Irr(H n (q)) bijectively with basically the same data.

Lusztig's reduction theorems.
In Paragraph 1.5 we already hinted at a link between affine Hecke algebras and graded Hecke algebras. The connected is established with two reduction steps, whose original versions are due to Lusztig [Lus3]. These simplifications do not work in the same way for all representations, they depend on the central characters. The first reduction step limits the set of central characters that has to be considered to understand all finite length H-modules.
By [Lus3,Lemma 3.15], for t ∈ T and α ∈ R: Fix t ∈ T . We will exhibit an algebra, almost a subalgebra of H, that captures the behaviour of H-representations with central character close to W t ∈ T /W . We consider the root system It fits in a root datum R t = (X, R t , Y, R ∨ t ), and λ and λ * restrict to parameter functions for R t . The affine Hecke algebra H(R t , λ, λ * , q) naturally embeds in H = H(R, λ, λ * , q).
But we have to be careful, this construction is not suitable when c α (t) for some α ∈ R. For instance, when R is of type A 1 and t = q = 1, the algebra H(R t , λ, λ * , q) is just C[X]. That is hardly helpful to describe all H-modules with weights in W q (e.g. ind H C[X] (C q ), triv and St). When c α (t) = 0 for all α ∈ R, H(R t , λ, λ * , q) detects most of what can happen with H-representations with central character W t, but still not everything.
Example 3.26. Consider the root datum R of type A 2 , with The point t ∈ T with t(e 1 − e 2 ) = t(e 2 − e 3 ) = e 2πi/3 satisfies R t = ∅ but W t = {id, (123), (132)}. We have H(R t , λ, λ * , q) = C[X], which possesses only one irreducible representation with central character t. On the other hand, the Rgroup for H and ξ = (∅, triv, t) is W t . Theorem 4.2 entails that ind H C[X] (C t ) splits into three inequivalent irreducible H-representations, all with central character W t.
The set is a root system, because λ and λ * are W -invariant. When t ∈ T un and (3.8) holds, R t coincides with R t because c α (t) cannot be 0. Let R + t = R + ∩ R t be the set of positive roots, and let ∆ t be the unique basis of R t contained in R + t . We warn that ∆ t need not be a subset of ∆. The group This is a group action of Γ t , so we can form the crossed product H(R t , λ, λ * , q) Γ t . We recall that this means H(R t , λ, λ * , q)⊗C[Γ t ] as vector spaces, with multiplication rule The group Γ t can be embedded in C(X) W ⊗ C[X] W H(R, λ, λ * , q) × with the elements ı • w from Proposition 1.12. In view of (3.11) and (3.12), this realizes H(R t , λ, λ * , q) Γ t as a subalgebra of C(X) W ⊗ C[X] W H(R, λ, λ * , q).
Unfortunately tensoring with C(X) W kills many interesting representations, so the above does not yet suffice to relate the module categories of H(R, λ, λ * , q), of H(R t , λ, λ * , q) Γ t and of a graded Hecke algebra. To achieve that, some technicalities are needed.
For an (analytically) open W -stable subset U of T , let C an (U ) be the algebra of complex analytic functions on U . As Z(H) injects in C an (U ) W , we can form the algebra . It can also be obtained from Definition 1.11 by using C an (U ) instead of O(T ), and as a vector space it is just C an (U ) ⊗ C H (W, q).
Let Mod f,U be the category of finite length H-modules, all whose O(T )-weights lie in U . and there is a well-defined group homomorphism This realizes H(R t , λ, λ * , q) Γ t as a subalgebra of H an (U ). Our version of Lusztig's first reduction theorem [Lus3,§8] is: Theorem 3.28. Assume that (3.8) holds and that t ∈ T satisfies W |t| −1 t = W t .
Then there exist open W t -stable neighborhoods U t of t with the following properties. (a) There is a natural embedding of C an (U ) W -algebras (a) and Proposition 3.27 induce equivalences of categories In that case the equivalences of categories between the outer terms in part (b) preserve temperedness and (essentially) discrete series.
Parts (a) and (b)  The conditions in Theorem 3.28 avoid possible unpleasantness caused by noninvertible intertwining operators. Algebras of the form H(R t , λ, λ * , q) Γ t behave almost the same as affine Hecke algebras. The difference can be handled with Clifford theory, as in [RaRa, Appendix]. In fact everything we said so far in this paper can be generalized to such crossed product algebras, see [Sol4,AMS3]. For simplicity, we prefer to keep the finite groups Γ t out of our presentation.
An advantage of Theorem 3.28 is that it reduces the study of Mod f (H) to those modules of H(R t , λ, λ * , q) Γ t all whose C[X]-weights belong to a small neighborhood of the point t, which is fixed by W (R t ) Γ t . We will use this in a loose sense, suppressing Γ t . Then Theorem 3.28 says that it suffices to consider those finite dimensional modules of an affine Hecke algebra H, all whose C[X]-weights lie in a small neighborhood of a W -fixed point t ∈ T .
The second reduction theorem will transfer such H-representations to representations of graded Hecke algebras. In the remainder of this paragraph we assume that u is fixed by W . We define a parameter function k u for the root system R u by (3.18) k u (α) = (λ(α) + α(u)λ * (α)) log(q)/2. By (3.16) α(u) ∈ {±1}, so k u is real-valued whenever the positivity condition (3.8) for q holds. This gives a graded Hecke algebra H(t, W, k u ). Let V ⊂ t be an analytically open W -stable subset. We can form the algebra Let Mod f,V (H(t, W, k u )) be the category of those finite dimensional H(t, W, k u )modules, all whose O(t)-weights lie in V . An analogue of Proposition 3.27 says that the inclusion H(t, W, k u ) → H(t, W, k u ) an (V ) induces an equivalence of categories The analytic map exp u : t → T, λ → u exp(λ) is W -equivariant, because u is fixed by W . It gives rise to algebra homomorphisms In good circumstances, the works already without involving rational (non-regular) functions.
Notice that the conditions in Theorem 3.29 always hold when V is a small neighborhood of 0 in t. When k u is real-valued, for instance whenever (3.8) holds, these condition also hold for V of the form a + V with V ⊂ ia a small ball around 0.
Combining Theorems 3.28, 3.29 and Proposition 3.27, we can draw important consequences for the category of finite dimensional H-modules 3.5. Analogues for graded Hecke algebras. Motivated by Corollary 3.30 we investigate all finite dimensional representations of graded Hecke algebras. The representation theory of graded Hecke algebras has been developed together with that of affine Hecke algebras, and they are very similar. As far as the topics in this section are concerned, these two kinds of algebras behave analogously. Instead of translating the paragraphs 3.1-4.1 to graded Hecke algebras, we will be more sketchy here, just providing the necessary definitions and pointing out the analogies. For more background and proofs we refer to [BaMo2,Eve,KrRa,Sol2,Sol3].
Compared to Paragraph 1.5, we assume in addition that W is a crystallographic Weyl group. Equivalently, the data t, a, R, W, S for a graded Hecke algebra now come from a based root datum R = (X, R, Y, R ∨ , ∆). Let k : R → C be a W -invariant parameter function and consider the algebra H = H(t, W, k).
A parabolic subalgebra of H is by definition of the form H P = H(t, W P , k) for a subset P ⊂ ∆. The associated "semisimple" quotient algebra is H P = H(t P , W P , k). The relation between these two subquotients of H is simple: In particular every irreducible representation of H P is of the form π P ⊗ λ for unique π P ∈ Irr(H P ) and λ ∈ t P . Parabolic induction for H is the functor ind H H P . Lemma 3.31. [AMS3, Theorems 2.5.b and 2.11.b] The equivalences of categories from Theorems 3.28 and 3.29 and Corollary 3.30 are compatible with parabolic induction.
Finite dimensional H-modules have weights with respect to the commutative subalgebra S(t * ) = O(t). With Theorem 3.29 one can translate many notions for H-modules to H-modules. In terms of weights, this boils down to replacing a subset U ⊂ T by exp −1 (U ) ⊂ t. More concretely, we say that a finite dimensional • essentially discrete series if Wt(π) ⊂ t ∆ + ia + a −− . Here V + a −− (for any V ⊂ t) is considered as empty when a −− = ∅.
Then a Langlands datum for H is a triple (P, τ, λ), where P ⊂ ∆, τ ∈ Irr(H P ) is tempered and λ ∈ ia P + a P ++ . Then ind H H P (τ ⊗ λ) is called a standard H-module. With these definitions, the Langlands classification (as in Theorem 3.13 and Lemma 3.15) holds true for graded Hecke algebras [Eve].
As expected, and proven in [AMS3,Theorem 2.11.d], the equivalence of categories between Mod f,V (H(t, W, k u )) and Mod f,exp u (V ) (H) resulting from Theorem 3.29 and Proposition 3.27 preserves temperedness and (essentially) discrete series. Combining that with Theorem 3.28, we find: Lemma 3.32. Assume that (3.8) holds. The equivalence of categories between Mod f,W u exp (a) (H) and Mod f,a (H(t, W (R u ), k u ) Γ u ) from Corollary 3.30 preserves temperedness and (essentially) discrete series.
From now on we assume that k : R → C has values in R, so that H is related to an affine Hecke algebra with parameters in R >0 . As induction data for H we take triplesξ = (P, δ, λ) where P ⊂ ∆, (δ, V δ ) ∈ Irr(H P ) is discrete series and λ ∈ t P . The space of such triples (with δ considered up to isomorphism) is denotedΞ. The parabolically induced representation attached to an induction datum is π(P, δ, λ) = ind H H P (δ ⊗ λ). Lemma 3.33. Letξ = (P, δ, λ) ∈Ξ.
(a) π(ξ) is tempered if and only if λ ∈ ia P . In that case π(ξ) is completely reducible. (b) Let W P cc(δ) be the central character of δ. The central character of π(ξ) is W (cc(δ) + λ). It lies in a/W if and only if λ ∈ a P .
Proof. The arguments from these statement will be given in the setting of affine Hecke algebras. From there they can be translated to graded Hecke algebras with Paragraph 3.4. (a) follows from Propositions 3.8 and 3.20.a. (b) The expression for the central character comes from Lemma 3.3. Since k is realvalued and δ is discrete series, W P cc(δ) lies in a P /W P [Slo2, Lemma 2.13]. These two facts imply the second statement.
The collection of intertwining operators between the representation π(ξ) with ξ ∈Ξ is simpler than for affine Hecke algebras, because t P ∩ t P = {0} does not contribute to it. There is a groupoid WΞ over ∆, with WΞ ,P P = {w ∈ W : w(P ) = P }.

Classification of irreducible representations
As in paragraph 3.3, we work with an affine Hecke algebra H = H(R, λ, λ * , q) where q > 1 and λ, λ * are real-valued. We abbreviate H = H(t, W, k) where W = W (R), t = Lie(T ) and k is a real-valued parameter function.
In Theorem 3.24 we reduced the classification of irreducible H-representations to a little combinatorics with a groupoid W Ξ and two substantial subproblems: • classify the irreducible discrete series representations δ of the parabolic subquotient algebras H P (modulo the action of T P ∩ T P via the automorphisms ψ P,u ), • determine the irreducible quotients of π(ξ) for ξ = (P, δ, t) ∈ Ξ + . In this section we address both these issues.

Analytic R-groups.
By Proposition 3.20.a-c the second subproblem above is equivalent to classifying the irreducible summands of the completely reducible H P (ξ) -representation . For that we have to analyse End H P (ξ) (π P (ξ) (ξ), which by Proposition 3.20.e boils down to investigating End H (π(ξ)).
For ξ = (P, δ, t) ∈ Ξ + we let W ξ be the subgroup of W Ξ,P P that stabilizes ξ (up to isomorphism of induction data). From Theorem 3.23 we know that the intertwining operators π(w, ξ) with w ∈ W ξ span End H (π(ξ)), but they need not be linearly independent. Knapp and Stein exhibited a subgroup R ξ of W ξ such that the π(w, ξ) with w ∈ R ξ do form a basis of End H (π(ξ)).
Let R + P be the set of positive roots in R P , with respect to the basis P . Suppose that α ∈ R + \ R + P and that P ∪ {α} is a basis of a parabolic root subsystem R P ∪{α} of R. Then we put We note that c P α is W P -invariant because W P stabilizes R P and does not make positive roots outside R P negative. Let δ ∈ Irr(H P ) be discrete series, with central character cc(δ) = W P r. Then t → c P α (rt) is a rational function on T P , independent of the choice of the representative r for cc(δ). For ξ = (P, δ, t) we consider the following subset of a P * : R ξ = ± α P : α ∈ R + \ R + P as above, c P α has a non-removable pole at rt . This generalizes the root system R t from (3.17).
Proposition 4.1. [DeOp2,Proposition 4.5] (a) R ξ is a reduced root system in a P * , The group W ξ acts naturally on a P * and stabilizes all the data used to construct R ξ , so it also acts naturally on R ξ . Clearly R + determines a set of positive roots R + ξ in R ξ . We define (4.1) R ξ = {w ∈ W ξ : w(R + ξ ) = R + ξ }. As W (R ξ ) acts simply transitively on the collection of positive systems of R ξ : Theorem 4.2. Let ξ ∈ Ξ + . (a) For w ∈ W ξ , the intertwining operator π(w, ξ) is scalar if and only if w ∈ W (R ξ ).

There are bijections
Proof. For ξ = (P, δ, t) with t ∈ T P un , all this (and more) was shown in [DeOp2, Theorems 5.4 and 5.5]. Using Proposition 3.20.a, the same proofs work for π P (ξ) (P, δ, t) with any ξ ∈ Ξ, they show the theorem on the level of H P (ξ) . Finally, we apply Proposition 3.20.c,e.
The last example shows that R-groups can be as complicated as W itself. This is in sharp contrast with the situations for real reductive groups and for classical p-adic groups, where all R-groups are abelian 2-groups. In all examples that we are aware of, the 2-cocycle ξ of R ξ is a coboundary, so that C[R ξ , ξ ] is isomorphic to C[R ξ ]. It would be interesting to know whether or not this is always true for affine Hecke algebras.
Although π(w, ξ) is only defined up to a scalar, the formula for ρ is independent of the choice of a normalization.
We denote this ρ by w(ρ), and we say that (ξ, ρ) and (w(ξ), w(ρ)) are W Ξassociate. It is not clear whether this comes from a groupoid action on a set containing all (ξ, ρ) as above, because W Ξ does not stabilizes the set of positive induction data Ξ + and we did not define R-groups for non-positive induction data.
Let us summarise some properties of standard H-modules.
Proof. (a) follows from Theorem 3.23. (b) is already contained in Theorem 4.2. (c) Recall from the remark after Theorem 3.24 that π determines a unique parabolically induced representation π(ξ), with ξ ∈ Ξ + , that has π as quotient. (d) is a consequence of parts (a), (b) and (c).
Corollary 4.4 provides a classification of Irr(H) in terms of induction data and R-groups. The role of the R-groups is quite subtle, firstly because it can be hard to determine them, secondly because potentially a non-trivial 2-cocycle can be involved in the H-endomorphism algebra of a parabolically induced representation.
Sometimes it is easier to work with standard modules than with irreducible representations, for their structure is more predictable.
• For H(R, q) with R of type A 1 , almost all standard modules are irreducible. The only reducible standard H-module is π(∅, triv, q) = ind H C[X] (C q ), which has triv as irreducible quotient. • For H n (q) all the groups R ξ are trivial, so the standard modules are just the parabolically induced representations π(ξ) with ξ ∈ Ξ + . • When H = H(R, 1) = C[X W ] (any R), there is a standard module π(∅, triv, t, ρ) for every t ∈ T and ρ ∈ Irr(W t ). In the notation from Paragraph 2.1 it equals π(t, ρ * ). In view of Theorem 2.2, these H-representations are irreducible and there are no other standard modules.

Residual cosets.
The most significant step towards the classification of discrete series Hrepresentations is the determination of their central characters, which was achieved by Opdam [Opd2]. Consider the following rational function on T : Its counterpart for H(t, W, k) is Definition 4.6. Let L ⊂ T be a coset of a complex algebraic subtorus of T . We call L a residual coset (with respect to R and q) if the zero order of c R along L is at least the (complex) codimension of L in T . An affine subspace l ⊂ a is called a residual subspace (with respect to R and k) if the zero order ofc R along l is at least the (real) codimension of l in a.
A residual point is a residual coset/subspace of dimension zero.
For any L or l as above, its zero order is always at most its codimension [Opd2,Corollary A.12]. Hence we may replace "is at least" by "equals" in Definition 4.6. This also implies that residual points can only exist if R spans a * . The collection of residual cosets/subspaces is stable under W , because R and q/k are so.
This point is the O(t)-character of the Steinberg representation of H, which by definition is onedimensional and restricts on C [W ] to the sign representation.
There are at most two W -orbits of residual points for H(C 2 , W (B 2 ), k), represented by (k 1 + k 2 , k 2 ) and (k 1 − k 2 , k 2 ). These points are indeed residual if The crucial property of residual points is: Theorem 4.8. [Opd2,Lemma 3.31] Let δ ∈ Irr(H) be discrete series. Then all its C[X]-weights are residual points for (R, q). Conversely, if t ∈ T is a residual point for (R, q), then there exists a discrete series H-representation with central character W t.
For q > 1, this is the unique C[X]-weight of the Steinberg representation of H(R, q)which is defined just like St for H n (q) in (2.11) and (2.13). Similarly ζ n t q is the C[X]-weight of the discrete series representation St⊗ζ n .
There is a general method to construct residual cosets from residual points for subquotient algebras. Namely, let P ⊂ ∆ and let r ∈ T P be a residual point for H P . Then T P r is a residual coset for H [Opd2,Proposition A.4]. Up to the action of W , every residual coset is of this form.
From that, Theorem 4.8 and Lemma 3.3 we deduce: for any induction datum (P, δ, t) ∈ Ξ, every weight of π(P, δ, t) lies in a residual coset of the same dimension as T P . Now we relate residual cosets to residual subspaces for graded Hecke algebras. By definition every residual coset for H can be written as L = u exp(λ)T L , where u ∈ T u n, λ ∈ a and T L ⊂ T is a complex algebraic subtorus.
Proposition 4.10. [Opd2,Theorem A.7] (a) With the above notations, λ + log |T L | ⊂ a is a residual subspace for H(t, W (R u ), k u ). (b) Every residual subspace for H(t, W (R u ), k u ) arises in this way. (c) When T L +T P and u exp(λ) is a residual point for H P , λ ∈ a P is a residual point for H(t P , W (R P,u ), k u ). In this case R P,u has the same rank as R P , namely |P |.
Example 4.11. Take X = Y = Z 2 , R = B 2 , R ∨ = C 2 = {±e 1 ± e 2 } ∪ {±2e 1 , ±2e 2 } and ∆ = {e 1 − e 2 , e 2 }. We write The points u ∈ T un with R u of rank 2 are (1, 1), (−1, −1), (1, −1) and (−1, 1), the last two being W -associate. There are at most 5 W -orbits of residual points, summarised in the following table: Here we give k u in terms of its values on a basis of R u . For generic parameters q 1 , q 2 , q 0 , all the five points of T in the above table are indeed residual, and each of them represents the central character of a unique discrete series representation [Sol4,§5.5].
As a consequence of Proposition 4.10, everything we said before about residual cosets for H can be translated to residual subspaces for graded Hecke algebras. Combining that with Lemma 3.32, we find that every O(t)-weight of a discrete series H-representation is a residual point in a.
In general it is easy to classify all points u ∈ T un for which R u has full rank in R, in terms of the affine Dynkin diagram of R [Opd2,Lemma A.8]. Recall that the relation between the representations of H(t, W (R u ), k u ) and of H(t, W (R u ), k u ) Γ u is well-understood from Clifford theory. Thus the classification of discrete series H-representation boils down to two tasks: • classify all residual points for (t, W, k), where k is any real-valued parameter function, • for a given residual point λ ∈ a, classify the discrete series H-representations with central character W λ.
In view of the isomorphism (1.25), it suffices to this when R is irreducible. The residual points for H(t, W, k) with such R and k have been classified completely in [HeOp,§4]. They are always linear expressions f (k) in the parameters k(α) for α ∈ R. For a given such f , f (k) is residual with respect to R and k for almost all k : R → R. We say that a parameter function k is generic if all potentially residual points f (k), for the H(t, W (R P ), k) with P ⊂ ∆, are really residual for this k, and are all different. From Theorem 4.13 one obtains a complete classification of discrete series representations of affine Hecke algebras with positive parameters. However, we have to point out that this does not yet achieve an actual classification of all irreducible representations. The problem is that it can remain difficult to effectively compute the R-groups R P,δ,t and their 2-cocycles P,δ,t from Paragraph 4.1.
We conclude this section with a discussion of the residual points for H in the most intricate case, for root systems of type B n . We take a = a * = R n , R = B n = {±e i : i =, 1 . . . , n} ∪ {±e i ± e j : i = j} and we write k(±e i ± e j ) = k 1 , k(±e i ) = k 2 . For every partition n = (n 1 , n 2 , . . . , n d ) of n we construct a point λ( n, k) ∈ a, or rather a S n -orbit in a, in the following way. Draw the Young diagram, with first column of n 1 boxes, second column of n 2 boxes and so on. Label the boxes from b 1 to b n in some way (to does not matter how, for another labelling will produce a point in the same S n -orbit in a). We define the height of a box b in column i and row j to be h(b) = j − i and we write For example, when n = 2 we have λ((2), k) = (k 1 + k 2 , k 2 ), λ ((1, 1), k) = (−k 1 + k 2 , k 2 ) Every residual point for H(C n , B n , k) is W (B n )-associate to a λ( n, k) [HeOp,Propositions 4.3 and 4.5]. For most parameters k i indeed all these points of a are residual, but not for all parameters. An extreme case is k 1 = k 2 = 0, then there are no residual points. A parameter function k : When k is generic all the λ( n, k) are residual, and they belong to different W (B n )orbits. On the other hand, when k is not generic, some of the λ( n, k) are not residual, and some of them may belong to the same W (B n )-orbit.

Geometric methods
We survey some of results on affine Hecke algebras obtained with methods from complex algebraic geometry. In many cases, these provide a complete classification of standard modules and of irreducible representations.

Equivariant K-theory.
In this paragraph we discuss equal label affine Hecke algebras, that is, with a single parameter q. Recall from Paragraph 1.2 that these algebras are especially important because they classify representations of reductive p-adic groups with vectors fixed by an Iwahori subgroup. It was realized independently by Kazhdan-Lusztig and by Ginzburg that such affine Hecke algebras can be realized as the equivariant Ktheory of a suitable complex algebraic variety. Then its representations can be analysed in algebro-geometric terms, and that leads to a beautiful construction and parametrization of all irreducible representations.
Let G be a connected complex reductive group with a maximal torus T , and let R(G, T ) be the associated root datum. We define H(G, T ) to be like H(R(G, T ), q), but with q replaced by an invertible formal variable z. As vector spaces where W = W (G, T ). The upcoming constructions work best when the derived group of G is simply connected, so we assume that in most of this paragraph. A main role is played by the Steinberg variety of G: The group G × C × acts on Z by (g, λ)(B, u, B ) = (gBg −1 , gu λ −1 g −1 , gB g −1 ).
Note that u λ −1 is defined because u is unipotent. This C × -action might appear ad hoc, but it is indispensable to obtain Hecke algebras. Without it, we could at best build the G-equivariant K-group K G (Z), which turns out be isomorphic to C[X] W [ChGi,Theorem 7.2]. According to [KaLu2,Theorem 3.5] and [ChGi,Theorem 7.2.5], there is a natural isomorphism The z's in H(G, T ) are due to the C × -action on Z. The ring of regular class functions on G is . When we regard z as the identity representation of C × , we can write the rings of regular class functions on C × and on G × C × as By construction [ChGi,§5.2], R(G×C × ) acts naturally on K G×C × (Z). On the other hand, a variation on (1.19) shows that R(G × C × ) is also naturally isomorphic to the centre of H(G, T ). With these identifications the isomorphism (5.1) is R(G × C × )linear. For any q ∈ C × we can specialize (5.1) to an isomorphism Further, let t ∈ G be a semisimple element and denote the associated onedimensional representation of R(G × C × ) by C t,q . Let Z t,q be the subvariety of Z fixed by (t, q) ∈ G × C × . By [ChGi,p. 414] there is an isomorphism The construction of K G×C × (Z)-modules is performed most naturally with Borel-Moore homology (that is equivalent to the constructions with equivariant K-theory in [KaLu2]). Let u ∈ G be a unipotent element such that tut −1 = u q . Let B be the variety of Borel subgroups of G and let B t,u be the subvariety of Borel subgroups that contain t and u. The convolution product in Borel-Moore homology [ChGi,Corollary 2.7.42] provides an action of H * (Z t,q , C) on H * (B t,u , C). This and (5.3) make H * (B t,u , C) into a K G×C × (Z)-module, usually reducible.
By [ChGi,Lemma 8.1.8] these constructions commute with the G-action, in the sense that intertwines the K G×C × (Z)-actions. In particular Z G (t, u) acts on H * (B t,u , C) by K G×C × (Z)-intertwiners. The neutral component of Z G (t, u) acts trivially, so we may regard it as an action of the component group π 0 (Z G (t, u)). That can be used to decompose the module H * (B t,u , C). Let ρ be an irreducible representation of π 0 (Z G (t, u)) which occurs in H * (B t,u , C). Then K t,u,ρ := Hom π 0 (Z G (t,u)) ρ, H * (B t,u , C) is a nonzero K G×C × (Z)-module, called standard in [KaLu2,5.12] and [ChGi,Definition 8.1.9]. Since the action factors via (5.3), K t,u,ρ can be regarded as a H(R(G, T ), q)-representation with central character W t. We call data (t, u, ρ) with the above properties a Kazhdan-Lusztig triple for (G, q).
This major result comes with a lot of extras. Firstly, suppose that L is a standard Levi subgroup of G and that it contains {t, u}. Then embeds naturally in H(R(G, T ), q) and, by [KaLu2,Theorem 6.2]: . For the second extra we suppose that q ∈ R >1 . By [KaLu2,Theorem 8.3] and [ABPS1, Proposition 9.3] the H(R(G, T ), q)-module L t,u,ρ is essentially discrete series if and only if {t, u} is not contained in any Levi subgroup of any proper parabolic subgroup of G.
With these two extras at hand, we can compare K t,u,ρ with the more analytic approach from Sections 3 and 4. Let L be a Levi subgroup of G which contains {t, u} and is minimal for that property. Upon conjugating everything by an element of G, we may assume that L is standard, that t ∈ T and that log |t| is as positive as possible in its W -orbit. Let ρ L ∈ Irr π 0 (Z L (t, u)) be an irreducible constituent of ρ| π 0 (Z L (t,u)) . Then (5.4) Hom π 0 (Z L (t,u)) ρ L , H * (B t,u L , C) is an irreducible essentially discrete series H(R(L, T ), q)-representation. By Theorem 3.18 it is of the form δ • ψ t , where δ is discrete series. Then ξ = (∆ L , δ, t ) ∈ Ξ + by the assumption on t. The induction of (5.4) to H(R(G, T ), q) contains K t,u,ρ as a direct summand. In view of Theorem 4.2.b, this summand must be picked out by an irreducible representation ρ of C[R ξ , ξ ]. We conclude that K t,u,ρ ∼ = π(ξ, ρ ), a standard module in the sense of Definition 3.12.
This argument also works in the opposite direction, and then it shows that the standard modules from Definition 3.12 are precisely the standard modules from [KaLu2] and [ChGi].
Recall that in Theorem 5.1 the complex reductive group G has simply connected derived group G der . Without that assumption on G, K G×C × behaves less well. Nevertheless, the parametrization of irreducible H(R(G, T ), q)-representations obtained in Theorem 5.1 is valid for any complex reductive group G. This was shown by Reeder [Ree2,Theorem 3.5.4], via reduction to the case with simply connected G der .
When q is a root of unity, Theorem 5.1 can definitely fail, in particular when q is a zero of the Poincaré polynomial of W . On the other hand, if q is not a zero of the polynomials (1.9) for any finite reflection subgroup of X W , then it seems likely that large parts of the K-theoretic approach are still valid.
A very special case arises when q = 1. Then Theorem 5.1 holds, in a slightly different, simpler way [Kat2]. In this case the action of H(R(G, T ), 1) = C[X * (T )] W on K t,u,ρ preserves the homological degree, so Hom π 0 (Z G (t,u)) ρ, H d (B t,u , C) is a subrepresentation for any d ∈ Z ≥0 . Then K t,u,ρ may obviously have many irreducible quotients, so the previous definition of L t,u,ρ cannot be used anymore. Instead we define that is, we only use the homology in the largest possible degree. It can be shown that (5.5) is the canonical irreducible quotient of K t,u,ρ [ABPS1,§12]. Thus modified, Theorem 5.1 becomes valid for q = 1 [Kat2,Theorem 4.1]. Notice that here the triples (t, u, ρ) satisfy tu = ut, so u is a unipotent element of Z G (t). This classification of Irr(X W ) can be regarded as a Springer correspondence for affine Weyl groups. The proof is much shorter than that of Theorem 5.1, it mainly relies on the Springer correspondence for finite Weyl groups.
As already observed in [KaLu2,§2], there are several equivalent ways to present Kazhdan-Lusztig parameters, when we consider them modulo G-conjugation (as in Theorem 5.1.c). One alternative shows the connection between different q's very nicely.
Let (t 1 , u, ρ 1 ) be a Kazhdan-Lusztig triple for (G, 1). Pick an algebraic homomorphism φ : SL 2 (C) → Z G (t 1 ) with φ(( 1 1 0 1 )) = u. By the Jacobson-Morozov theorem, such a φ exists and is unique up to conjugation by Z G (t 1 , u). Assume that we have a preferred square root of q. We put t q = t 1 φ( q 1/2 0 0 q −1/2 ), so that t q ut −1 q = u q . Then B t 1 ,u and B tq,u are homotopy equivalent and ρ 1 gives rise to a unique ρ q [ABPS1, Lemma 6.1]. This provides a bijection between G-conjugacy classes of Kazhdan-Lusztig tripes for (G, 1) and for (G, q) [ABPS1, Lemma 7.1]. In combination with Theorem 5.1 we obtain: Corollary 5.2. Let q ∈ C × be either 1 or not a root of unity. There exists a canonical bijection {Kazhdan-Lusztig triples for (G, 1)}/G ←→ Irr H(R(G, T ), q) (t 1 , u, ρ 1 ) → L tq,u,ρq One advantage of this parametrization is that the pair (t 1 , u) is the Jordan decomposition of an arbitrary element of G, so that (t 1 , u) up to G-conjugacy parametrizes the conjugacy classes of G.
Example 5.3. Let R be of type GL n and consider H n (q). As Z GLn(C) (t 1 , u) = Z GLn(C) (t 1 u) is always connected, ρ is necessarily trivial and may be ignored. Corollary 5.2 recovers the parametrization of Irr(H n (q)) summarised in Theorem 2.6.
In addition to the already discussed properties of these bijections, we mention that temperedness can be detected easily in Corollary 5.2. Namely, by [ABPS1,Proposition 9.3], for q ∈ R ≥1 : (5.6) L tq,u,ρq is tempered ⇐⇒ t 1 lies in a compact subgroup of G.

Equivariant homology.
The material in the previous paragraph learns us a lot about equal label affine Hecke algebras, but very little about the cases with several parameters q s . It appears that equivariant K-theory is not flexible enough to incorporate more than one independent q-variable.
Faced with this problem, Lusztig discovered that Hecke algebras with multiple parameters can still be studied geometrically, if we accept two substantial modifications: • replace affine Hecke algebras by their graded versions, • replace equivariant K-theory by equivariant homology.
Not all combinations of q-parameters can obtained in this way, but still a considerable number of them. Our treatment of this method is based on the papers [Lus1,Lus2,Lus4,Lus6,AMS2].
Let G be a connected complex reductive group and let P be a parabolic subgroup of G with Levi factor L and unipotent radical U . We denote the Lie algebras of these groups by g, p, l and u. Let v ∈ l be nilpotent and let C L v be its adjoint orbit. Let L be an irreducible L-equivariant cuspidal local system on C L v . We refrain from explaining these notions here, instead we refer to [Lus1], where cuspidal local systems are introduced and classified.
To the above data we will associate a graded Hecke algebra. We take T = Z(L) • , a (not necessarily maximal) torus in G. According to [Lus2,Proposition 2.5], the cuspidality of L implies that: • the set of weights of T acting on g is a (possibly nonreduced) root system The parabolic subgroup P determines a basis ∆ L of R(G, T ). Let t = X * (T ) ⊗ Z C be the Lie algebra of T , so that The action of W L on T induces actions on t and t * , which stabilize R(G, T ). The definition of the parameter function k : R(G, T ) → Z involves the nilpotent element v. Let g α ⊂ g be the root space and let s α ∈ W L be the reflection associated to α ∈ R(G, T ). Since v ∈ l commutes with t, ad(v) stabilizes each g α . For α ∈ ∆ L one defines k(α) ∈ Z ≥2 by Then k(α) = k(β) whenever α, β ∈ ∆ L are W L -associate. Now we can define H(G, L, L, r) = H(t, W L , k, r).
Suppose that G is an almost direct product of connected normal subgroups G 1 and G 2 . Then L, C L v and L decompose accordingly and H(G, L, L, r) = H(G 1 , L 1 , L 1 , r) ⊗ C[r] H(G 2 , L 2 , L 2 , r).
If G is a torus, then necessarily L = T = G and v = 0. In that case L is trivial and H(T, T, L, r) is just O(t) ⊗ C C [r]. Hence the study of H(G, L, L) can be reduced to simple G. Then it becomes feasible to classify the data, and indeed this has been done in [Lus2,2.13]. We tabulate the possibilities for g, l, R(G, T ) and k: In this table d, p ∈ Z >0 are arbitrary, α ∈ R(G, T ) is a long root and β ∈ R(G, T ) is a short root. (For R(G, T ) of type BC d we mean that α = ±e i ± e j and β = ±e i .) Recall from (1.22) that we can simultaneously rescale all the k(α) without changing the algebra (up to isomorphism). When R(G, T ) has roots of different lengths, we can also adjust the parameters for roots of one length in a specific way: Example 5.4. Let R be of type B n , F 4 or G 2 . Assume that R spans a * . Any parameter function k for R has two independent values k 1 = k(α) and k 2 = k(β), which can be chosen arbitrarily. We write k = (k 1 , k 2 ) and we consider the graded Hecke algebra H(t, W (R), k 1 , k 2 , r). Take = 2 for B n or F 4 , and = 3 for G 2 . The set is a root system in a * , of type (respectively) C n , F 4 or G 2 . Notice that now α is short and β is long. The identity map on the vector space underlying H(t, W (R), k 1 , k 2 , r) provides an algebra isomorphism H(C n , W (B n ), k 1 , k 2 , r) → H(C n , W (C n ), 2k 2 , k 1 , r) (5.8) H(C 4 , W (F 4 ), k 1 , k 2 , r) → H(C 4 , W (F 4 ), 2k 2 , k 1 , r) (5.9) H(C 2 , W (G 2 ), k 1 , k 2 , r) → H(C 2 , W (G 2 ), 3k 2 , k 1 , r) (5.10) In particular any graded Hecke algebra of type C n is also a graded Hecke algebra of type B n (but with different parameters).
We will call a parameter function obtained from the above table by a composition of the isomorphisms (1.22) and those from Example 5.4 geometric. Thus we have a large supply of geometric parameter functions for type B/C root systems.
Next we describe the geometric realization of H(G, L, L, r). We need the varietieṡ The first is a variation on the variety B of Borel subgroups of G, while the second has a flavour of the Steinberg variety of G. The group G × C × acts onġ by (g 1 , λ)(x, gP ) = (λ −2 Ad(g 1 )x, g 1 gP ), and similarly ong N . The L × C × -equivariant local system L on C L v yields a G × C × -equivariant local systemL onġ. The two projectionsg N →ġ give rise to an equivariant local systemL =L L * ong N . Equivariant (co)homology with coefficients in a local system is defined in [Lus2,§1]. In special cases this theory admits a convolution product [Lus4,§2]. That makes H G×C × * (g N ,L) into a graded algebra, see the proof of [Lus4,Theorem 8.11].
Theorem 5.5. [Lus2,Corollary 6.4] and [Lus4,Theorem 8.11] There exists a canonical isomorphism of graded algebras With equivariant homology one can construct many modules for H G×C × * (g N ,L). Let y ∈ g be nilpotent and define P y = {gP ∈ G/P : Ad(g −1 )y ∈ C L v + u}. This is the appropriate analogue of the variety B u of Borel subgroups containing u. The group M (y) := {(g 1 , λ) ∈ G × C × : Ad(g 1 )y = λ 2 y} acts on P y by (g 1 , λ)gP = g 1 gP . The inclusion {y} × P y →ġ is M (y)-equivariant, which allows us to restrictL to an equivariant local system on P y . With constructions in equivariant (co) (P y ,L), the restriction of E y,σ,r to C[W L ] does not depend on (σ, r). As usual, the isomorphism class of E y,σ,r depends only on (y, σ, r) up to the adjoint action of G (which fixes r). From the action of π 0 (M (y)) on H M (y) • * (P y ,L), only the operators that stabilize the adjoint orbit [σ] of σ in Lie(M (y) • ) act on E y,σ,r . Hence irreducible representations ρ of π 0 (M ) [σ] can be used to decompose the H(G, L, L, r)-module E y,σ,r further. We define the H(G, L, L, r)representation E y,σ,r,ρ = Hom π 0 (M (y)) [σ] (ρ, E y,σ,r ).
In the previous paragraph we encountered a clear condition on the representation ρ of the component group: it should appear in the homology of a particular variety, otherwise the associated module would be 0.
In the current setting the condition on ρ is more subtle, because P y can be empty and a local system L is involved. To formulate it we need the cuspidal support map Ψ G from [Lus1,6.4]. It associates a cuspidal support (L , C L v , L ) to every pair (x, ρ ) with x nilpotent and ρ ∈ Irr π 0 (Z G (x)) . Giving such (x, ρ) is equivalent to giving a G-equivariant cuspidal local system on a nilpotent orbit in g (which is also an equivariant perverse sheaf). The cuspidal support map can be expressed with a version of parabolic induction for equivariant perserve sheaves [Aub,§4.1]. According to [AMS2,Proposition 3.7]: . When L = T is a maximal torus of G and v = 0, we have P y = B exp y and L is trivial. Then the condition (5.13) reduces to: ρ appears in H * (B exp y Z G (σ 0 ) , C). That is equivalent to the condition on ρ in the Kazhdan-Lusztig triple (exp(σ 0 ), exp(y), ρ) for (G, 1).
(a) When (r) ≥ 0, the following are equivalent: • E y,σ 0 ,r,ρ is tempered, When (r) > 0, the following are equivalent: • M y,σ 0 ,r,ρ is essentially discrete series, • y is distinguished nilpotent in g, that is, not contained in any proper Levi subalgebra of g. Moreover in this case σ 0 ∈ Z(g) and E y,σ 0 ,r,ρ = M y,σ 0 ,r,ρ . (c) When r ∈ R, the central character of E y,σ 0 ,r,ρ lies in a/W if and only if σ 0 ∈ Ad(G)a.
Proof. (a) and (b) See [AMS2,(84) and (85)]. (c) Upon conjugating the parameters by a suitable element of G, we may assume that σ 0 , dγ y AMS2,Proposition 3.5.c]. Then dγ y r 0 0 −r represents the central character of the module E y,0,r = IM * E y,dγy r 0 0 −r ,r for Part (b) tells us that the restriction of E y,0,r to H(Z G (σ 0 ) der , L ∩ Z G (σ 0 ) der , L, r) is a direct sum of discrete series representations. By [Slo2,Lemma 2.13] the central characters of these representations lie in Z g (σ 0 ) der ∩a/W L∩Z G (σ) der . Hence dγ y r 0 0 −r ∈ a, which implies that σ 0 ∈ a ⇐⇒ σ 0 − dγ y r 0 0 −r ∈ a. Compare that with (5.15) The family of representations E y,σ,r is compatible with parabolic induction, under a mild condition that (σ, r) is not a zero of a certain polynomial function [Lus6,Corollary 1.18]. Namely, let Q be a standard Levi subgroup of G containing L and suppose that {y, σ 0 } ⊂ Lie(Q). When (σ, r) = 0 (or r = 0, see [AMS2,Theorem A.2]), there is a canonical isomorphism The Iwahori-Matsumoto involution commutes with parabolic induction, so (5.16) also holds for the family of representations E y,σ 0 ,r,ρ . With that and Theorem 5.6 one can show that (at least when (r) > 0) E y,σ 0 ,r,ρ is a standard module in the sense of Paragraph 3.5, see [AMS2,Proposition A.3]. (This refers to the last arXiv-version of [AMS2], in which an appendix was added to deal with a mistake in the published version.) The argument for standardness is analogous to what we sketched for K t,u,ρ around (5.4).
Example 5.8. We illustrate the material in this section with an example of rank 1. Let G = Sp 4 (C), L = Sp 2 (C)×GL 1 (C) and v regular nilpotent in l. The local system on C L v corresponding to the sign representation of π 0 (Z L (v)) = π 0 (Z(L)) = {±1} is cuspidal. The root system with respect to T = Z(L) • ∼ = C × is R(G, T ) = {±α, ±2α}, of type BC 1 . Its Weyl group is W L = {1, s α }, with s α acting on T by inversion. The parameter function by determined by L satisfies k(±α) = 3 and k(±2α) = 6. The associated graded Hecke algebra is H(G, L, L, r) = H(t, W L , k, r). The irreducible representations on which r acts by a fixed r ∈ C × were classified in Example 3.34 and for r = 0 in Paragraph 2.2.
Let us analyse the possibilities for the geometric parameters. It turns out that the condition on ρ can only be met for nilpotent elements y in two adjoint orbits in g: the orbits of v and of v + v , where v is regular nilpotent in Z(l der ) ∼ = sp 2 (C). The cuspidal support condition becomes that the subgroup of π 0 (Z G (σ 0 , y)) coming from π 0 (Z(L)) must act via the sign representation.
We may assume that Cv is stable under the adjoint action of T . A complete list of representatives for the G-conjugacy classes of parameters for H(G, L, L, r) is: O(t)⊗ C C[r] (C diag(σ,r,−r,−σ),r ) is irreducible, and hence equal to both M y,σr,r,ρ and M y,σr,r,ρ . The exceptions are σ = ±r, then M y,diag(r,r,−r,−r),r,ρ = St and M y,diag(r,r,−r,−r),r,ρ = triv.

Affine Hecke algebras from cuspidal local systems.
In the previous paragraph we associated a graded Hecke algebra to a cuspidal local system on a nilpotent orbit in a Levi algebra of g. With a process that is more or less inverse to the reduction theorems in Paragraph 3.4, we can glue a suitable family of such algebras into one affine Hecke algebra.
Concretely, let G, P, L, v, T, L be as before. Let R(G, T ) ∨ red ⊂ X * (T ) be the dual of the reduced root system R(G, T ) red ⊂ X * (T ), and consider the root datum red , X * (T ), ∆ L ). There are unique parameter functions λ, λ * : R(G, T ) red → Z ≥0 that meet the requirements sketched above, see [AMS3,Proposition 2.1 and (28)]. With those, we define This is slightly different from [AMS3,§2], where the algebras involved a formal variable z instead of q 1/2 . To compensate for the square root (q 1/2 versus q) we could replace λ, λ * by 2λ, 2λ * -but that is not necessary, since we are at liberty to choose q ∈ R >0 as we like.
When λ, λ * : R → R arise from a cuspidal local system as in (5.17), we call them geometric parameter functions for R. As we may choose any q ∈ R >0 , λ, λ * may be scaled by any nonzero real factor, and remain geometric.
For a unitary t ∈ T un , we saw in Corollary 3.30 that there is an equivalence of categories Let Z G (t) be the subgroup of G generated by Z G (t) and the root subgroups U α with s α (t) = t. By [AMS3, Theorems 2.5 and 2.9] we may identify Here Z G (t) has component group Γ t , so it can be disconnected. In that case these graded Hecke algebras are a little more general than in the previous paragraph, but that does not matter much.
With Theorems 3.28 and 3.29 we can reduce the classification of Irr(H(G, L, L)) to Theorem 5.6. For better additional benefits, we prefer to use the modules E y,σ 0 ,r,ρ and M y,σ 0 ,r,ρ . In this context it is more natural to use data from G that from g. The correct parameters turn out to be a variation on Kazhdan-Lusztig triples, namely triples (s, u, ρ) where , exp * (L)) modulo G-conjugacy. When Z G (s) is disconnected, we have to use a generalization of the cuspidal support map, defined in [AMS1,§4]. By conjugating with a suitable element of G, we may always assume that s ∈ T . Let E s,u,ρ (resp. M s,u,ρ ) be the H(G, L, L)-module obtained from E log u,log |s|,log(q)/2,ρ ∈ Mod H( Z G (s|s| −1 ), L, L, r)/(r − log(q)/2) (resp. M log u,log |s|,log(q)/2,ρ ) via Theorems 3.29 and 3.28, with respect to s|s| −1 ∈ T .
Theorem 5.10. [AMS3,Theorem 2.11] Let q ∈ R >0 and consider triples (s, u, ρ) as above. (b) Suppose that s ∈ T and let γ u : SL 2 (C) → Z G (s) be an algebraic homomorphism with γ u (( 1 1 0 1 )) = u. Then E s,u,ρ and M s,u,ρ admit the central character W L sγ u ( q 1/2 0 0 q −1/2 ). (c) Suppose that q ≥ 1. The following are equivalent: • E s,u,ρ is tempered, • M s,u,ρ is tempered, • s lies in a compact subgroup of G. (d ) Suppose that q > 1. Then M s,u,ρ is essentially discrete series if and only if u is distinguished unipotent in G (that is, not contained in any proper Levi subgroup of G).
Notice that q = 1 is allowed here. For q = 1, Theorem 5.10 provides a parametrization of Irr(X * (T ) W (G, T )) with G-association classes of triples (s, u, ρ) as above. That can be regarded as an affine version of the generalized Springer correspondence from [Lus1].
Example 5.11. We work out the parametrization from Theorem 5.10 for (G, L, L) as in Examples 5.8 and 5.9. Here H(G, L, L) is of type A 1 , with parameters λ(α) = 4 and λ * (α) = 2. With the example at the end of Paragraph 5.2 at hand, it is easy to determine all geometric parameters (s, u, ρ) for H(G, L, L), and the relevant modules can be found by applying Theorems 3.29 and 3.28.
A comparison with Paragraph 2.3 shows that we indeed found every irreducible H(G, L, L)-representation once in this way.
We have associated to every complex reductive group G a family of affine Hecke algebras, one for every cuspidal local system on a nilpotent orbit for a Levi subgroup of G. Every nilpotent orbit admits only a few inequivalent cuspidal local systems, and G-conjugate data (L, C L v , L) yield isomorphic Hecke algebras. Thus we have a finite family of affine Hecke algebras associated to G.
For this algebra the cuspidal support condition on the triples (s, u, ρ) reduces to the condition on ρ in a Kazhdan-Lusztig triple for (G, 1). By [AMS3,Proposition 2.18], the parametrization of standard and irreducible H(G, T, L = triv)-modules in Theorem 5.10 coincides with the Kazhdan-Lusztig paramerization from Theorem 5.1, modified as in Corollary 5.2. That is less obvious than it might seem though, the twist with the Iwahori-Matsumoto involution in (5.14) is necessary to achieve the agreement.

Comparison between different q-parameters
The aim of this section is a canonical bijection between the set of irreducible representations of an affine Hecke algebra with arbitrary parameters q s ∈ R ≥1 , and the set of irreducible representations of the same algebra with parameters q s = 1. This will be achieved in several steps of increasing generality.
Consider any graded Hecke algebra H = H(t, W, k). The group algebra C [W ] is embedded in H, so every H-representation can be restricted to a W -representation. For k = 0, the isomorphism shows that a representation on which O(t) acts via evaluation at 0 ∈ t is the same as a C[W ]-representation. From Example 3.7 we know that the irreducible tempered representations of H(R, 1) with central character in exp (a) are precisely the irreducible representations which admit the O(T )-character 1 ∈ T . Via Corollary 3.30 this implies that the irreducible representations of (6.1) are precisely the irreducible tempered H(t, W, 0)-representations whose central character is real, that is, lies in a/W .
That and the results of Paragraph 3.4 indicate that we should focus on Hrepresentations with O(t)-weights in a = X * (T ) ⊗ Z R. We say that those have real weights. Let Irr 0 (H(t, W, k)) = {π ∈ Mod f,a (H(t, W, k)) : π is irreducible and tempered} be the set of irreducible tempered representations with real central character. The above says that Irr 0 (H(t, W, 0)) can be identified with Irr (W ). Since H(t, W, k) is a deformation of H(t, W, 0), it can be expected that something similar holds for H(t, W, k). On closer inspection the parameters k(α) interact with the notion of temperedness, and it is natural to require that k takes real values.
For later applications to affine Hecke algebras, it will pay off to increase our generality. Let Γ be a finite group which acts on (a * , z * , R, ∆). That is, Γ acts Rlinearly on a * , and that action stabilizes R, ∆ and the decomposition a * = RR ⊕ z * . Suppose further that k : R → R is constant on Γ-orbits. Then Γ acts on H(t, W, k) by the algebra automorphisms The crossed product algebra H(t, W, k) Γ = H Γ is of the kind already encountered in Corollary 3.30. The Γ-action on H preserves all the available structure, so all the usual notions for H also make sense for H Γ.
We denote the restriction of any H Γ-representation π to the subalgebra C[W Γ] by Res W Γ (π). An initial result in the direction sketched above is: Theorem 6.1. [Sol2,Theorem 6.5] Irr 0 (H Γ) and Irr(W Γ) have the same cardinality, and the set Res W Γ (Irr 0 (H Γ)) is linearly independent in the representation ring of W Γ.
In particular, it is possible to choose a bijection Irr 0 (H Γ) → Irr(W Γ) such that the image of π ∈ Irr 0 (H Γ) is always a constituent of Res W Γ (π). We will establish a much more precise version of Theorem 6.1, for almost all positive parameter functions k.
Theorem 6.2. Let k : R → R ≥0 be a Γ-invariant parameter function whose restriction to any type F 4 component of R is geometric or has k(α) = 0 for a root α in that component. (a) The set Res W Γ (Irr 0 (H Γ)) is a Z-basis of Z Irr(W Γ). (b) There exist total orders on Irr 0 (H Γ) and on Irr(W Γ) such that the matrix of the Z-linear map is upper triangular and unipotent. (c) There exists a unique bijection ζ H Γ : Irr 0 (H Γ) → Irr (W Γ) such that, for any π ∈ Irr 0 (H Γ), ζ H Γ (π) occurs in Res W Γ (π).
Recall from Table (5.7) that the geometric parameter functions for R = F 4 are given by (k(α), k(β)) equal to (k, k), (2k, k), (k, 2k) or (4k, k) for any k ∈ C × , where α is a short root and β is a long root. We expect that Theorem 6.2.b also holds for non-geometric parameter functions k : F 4 → R >0 , but found it too cumbersome to check.
The discussion after (6.1) shows that the theorem is trivial for k = 0, so we assume from now on that k = 0. Our proof of Theorem 6.2 will occupy the entire paragraph. Lemma 6.3. Theorem 6.2 holds for H(t, W, k) with k : R → R >0 geometric.
Proof. The only irreducible tempered representation with real central character of H(t, R = ∅, k) = O(t) is C 0 , so the result is trivially true for that algebra. In view of the decomposition of H according to the irreducible components of R (1.25), we may assume that R is irreducible and spans a * .
The algebra isomorphisms (5.8)-(5.10) are the identity on O(t), so they preserve the set Irr 0 (H). The same holds for m z from (1.22), when z ∈ R >0 . Therefore we may just as well suppose that k is one of the parameter functions in the table (5.7), and that H = H(G, L, L, r)/(r − r) for some r ∈ R >0 . By Theorem 5.7 Irr 0 (H) consists of the representations M y,0,r,ρ with y ∈ g nilpotent, σ 0 = 0, ρ ∈ Irr π 0 (Z G (y)) and Ψ G (y, ρ) = (L, C L v , L) up to G-conjugacy. By Lemma 3.33 all the irreducible constituents of E y,0,r,ρ are tempered and have central character in a/W . For r = 0 all these representations admit the O(t)-character 0, so they can be identified with W -representations via (6.1).
Theorem 5.6.d for r = 0 says that all constituents of E y,0,0,ρ different from M y,0,0,ρ are of the form M y ,0,0,ρ with dim C G y < dim C G y . The numbers dim C G y define a partial order on the set of eligible pairs (y, ρ), considered modulo G-conjugation. Refine that to a total order. We transfer that to a total ordering on Irr 0 (H) (resp. Irr(W )) via (y, ρ) → M y,0,r,ρ (resp. M y,0,0,ρ ). With respect to these orders, the matrix of Res W : Z Irr 0 (H) → Z Irr(W ) is unipotent and upper triangular. It follows that ζ H : M y,0,r,ρ → M y,0,0,ρ is the unique map Irr 0 (H) → Irr(W ) with the required properties.
We remark that Lemma 6.3 with respect to Lusztig's alternative version of temperedness was proven in [Ciu,§3].
When R is irreducible and all roots have the same length, k is determined by the single number k(α) > 0, and it is geometric. So we just dealt with R of type A n , D n , E 6 , E 7 or E 8 . For R of type B n , C n , F 4 or G 2 , let k 1 be the k-parameter of a long root and k 2 the k-parameter of a short root. We will consider the algebras H(t, W (R), k) with k 1 = 0 or k 2 = 0 in and after (6.11).
That settles R = F 4 for the moment, because we excluded non-geometric strictly positive parameters. For R = G 2 with strictly positive k, Theorem 6.2 was proven in [Slo1,§1.4.4], by working it out completely for all possible cases.
Notice that all k with k 2 /k 1 ∈ (p − 1/2, p) are generic for B n and for all its parabolic root subsystems. Hence the residual subspaces of a for k (Definition 4.6) are canonically in bijection with those for k . More precisely, every residual subspace has coordinates that are linear functions of k, see Proposition 4.10 and (4.7). If such a linear function gives a residual subspace for k , then it also gives a residual subspace for k, and conversely. From Theorem 4.13 we obtain a canonical bijection between the sets of irreducible discrete series representations of H P (C n , W (B n ), k ) and of H P (C n , W (B n ), k), say δ → δ, where the image depends continuously on k.
By Lemma 3.33 and (4.5), Irr 0 (H(C n , W (B n ), k)) consists of the representations π(P, δ, 0, ρ) with ρ ∈ Irr(C[R P,δ,0 , P,δ,0 ]). Recall from Paragraph 4.1 that the analytic R-group R P,δ,λ is defined in terms of the functionsc α (λ). Since k is only allowed to vary among generic parameters, the pole order ofc α | a P * at λ = 0 does not depend on k. Hence R P,δ,0 does not depend on k either. The intertwining operators π(w, P, δ, 0) (w ∈ R P,δ,0 ) that span the twisted group algebra can be constructed so that they depend continuously on k. Then C[R P,δ,0 , P,δ,0 ]) becomes a continuous family of finite dimensional semisimple C-algebras. Such algebras cannot be deformed continuously, so the family is isomorphic to a constant family. That provides a canonical bijection Irr(C[R P,δ ,0 , P,δ ,0 ]) → Irr(C[R P,δ,0 , P,δ,0 ]), We plug it into (4.5) and we obtain a bijection where the image depends continuously on k. Finite dimensional representations of the finite group W do not admit continuous deformations, so (6.2) preserves Wtypes. Knowing that, Theorem 6.2 for H(C n , W (B n ), k ), as shown in Lemma 6.3, immediately implies Theorem 6.2 for H(C n , W (B n ), k).
Now we involve the group Γ that acts on H via automorphisms of (a * , z * , R, ∆).
Lemma 6.5. Let H be one of the graded Hecke algebras for which we already proved Theorem 6.2. We can choose the total orders on Irr 0 (H) and on Irr(W ) such that, for any π ∈ Irr 0 (H) and any irreducible constituent π of Res W (π) different from ζ H (π): γ * (π ) > ζ H (π) for all γ ∈ Γ.
Proof. First we assume that R is irreducible and spans a * . We consider a geometric k and we revisit the proof of Lemma 6.3. Replacing H by an isomorphic algebra, we may assume that k comes from table (5.7). Inspection of the table shows that every automorphism of (R, ∆) can be lifted to an automorphism of (G, T ) (Recall that there are no automorphisms of (R, ∆) when R has type B d , BC d , G 2 or F 4 , apart from the identity.) Hence the function Irr(W ) → R : M y,0,0,ρ → dim C G y is Γ-invariant. We replace this function by a function f R,k : Irr(W ) → R, whose images differ only slightly from dim C G y and which induces the total order on Irr(W ) claimed in Theorem 6.2.b and exhibited in the proof of Lemma 6.3. Via ζ H , we also regard f R,k as a function Irr 0 (H). Then Theorem 6.2.c implies that every constituent of Res W ( M y,0,r,ρ ) different from M y,0,0,ρ is isomorphic to a M y ,0,0,ρ with (6.3) f R,k γ * ( M y ,0,0,ρ ) > f R,k ( M y,0,0,ρ ) for all γ ∈ Γ.
When R = G 2 and k is not geometric, we can use the analysis from [Slo1,§1.4.4] to find a function f R,k : Irr(W ) → R with analogous properties. For other nongeometric k, we may assume that R = B n (recall we imposed that k is geometric for R = F 4 ). Then k is one of the parameter functions considered in Lemma 6.4. In the proof of that lemma we saw that k can be deformed continuously to a geometric parameter function k , while staying generic. That led to a canonical bijection (6.4) Irr 0 (H) → Irr 0 (H(C n , W (B n ), k )), which preserves W -types. We define f R,k to be the composition of f R,k with that bijection, and we transfer it to a function on Irr(W ) via ζ H . The bijection (6.4) and ζ H are Γ-equivariant, if nothing else because (B n , ∆) does not admit nontrivial automorphisms. Hence all the properties of f R,k transfer to f R,k . So far we proved the lemma in all cases where R is irreducible and we already had Theorem 6.2, and we made Theorem 6.2.b more explicit by associating the total order to a real-valued function f R,k . For a general R we use the decomposition (1.25) of H. It provides a natural bijection . We may assume that all values of the f R i ,k constructed above are algebraically independent and differ from an integer by at most (8d) −1 (if not, we can adjust them a bit). Now we define f R,k : Irr 0 (H) → R by and we order Irr 0 (H) accordingly. Via ζ H we transfer f R,k and the total order to Irr (W ). Let π be a constituent of and let π i ∈ Irr(W (R i )), i = 1, . . . , d be its tensor components. At least one of the π i is not isomorphic to ζ H(t i ,W (R i ),k) (V i ). By (6.3) and its analogues for other irreducible R: (6.5) f R,k (γ * (π )) > f R,k (π) + 1 − 2d/8d = f R,k (ζ H (π)) + 3/4 for all γ ∈ Γ.
The restriction of π σ to H has constituents γ * (π) for γ ∈ Γ/Γ π , each appearing with multiplicity dim V σ . Since Γ stabilizes a, π σ has all O(t)-weights in a if and only if that holds for π. As Γ stabilizes ∆, it preserves temperedness of Hrepresentations. Consequently π σ is tempered if and only if π is tempered. In particular Irr 0 (H Γ) consists of the representations π σ with π ∈ Irr 0 (H) and σ ∈ Irr(C[Γ π , π ]). Lemma 6.6. Let H be one of the graded Hecke algebras for which we already proved Theorem 6.2 and Lemma 6.5. Then Theorem 6.2 holds for H Γ.
Take a total order on Irr 0 (H Γ) that refines the partial order defined by f R,k . We transfer f R,k and this total order to Irr(W Γ) via the bijection (6.8). Then the above verifies parts (a) and (b) of Theorem 6.2 for H Γ. It follows this there is a unique ζ H Γ that fulfills the requirements, namely (6.8) Notice that in Lemma 6.4 we did not allow k 2 /k 1 ∈ (0, 1/2). Other special cases that we skipped in Lemmas 6.3 and 6.4 are: (6.11) H(C n , W (B n ), k) = H(C n , W (D n ), k 1 ) s en when k 2 = 0, H(C n , W (B n ), k) = H(C n , W (A 1 ) n , k 2 ) S n when k 1 = 0, H(C 4 , W (F 4 ), k) ∼ = H(C 4 , W (D 4 ), k i ) S 3 when k 3−i = 0, H(C 2 , W (G 2 ), k) ∼ = H(C 2 , W (A 2 ), k i ) S 2 when k 3−i = 0.
Let us relate some of these cases.

There exists a canonical bijection
Irr 0 H(C n , W (B n ), k) → Irr 0 H(C n , W (D n ), k 1 ) s en which preserves W (B n )-types.
Proof. We abbreviate H = H(C n , W (B n ), k) and H = H(C n , W (D n ), k 1 ) s en . As explained in the proof of Lemma 6.4, for k within the range of parameters considered in this lemma, Irr 0 (H) is essentially independent of k. By varying k 2 continuously, we can reach the algebra H , which however may behave differently. By Clifford theory Irr 0 (H ) consists of representations of the following kinds: (i) ind H H(C n ,W (Dn),k 1 ) (π), where π ∈ Irr 0 (H(C n , W (D n ), k 1 )) is not equivalent with s * en (π), (ii) V π ⊗V σ , where π ∈ Irr 0 (H(C n , W (D n ), k 1 )) is fixed by s * en and σ ∈ Irr( s en ) = {triv, sign}.
Let us investigate what happens when we deform k 2 = 0 to a positive but very small real number. Accordingly we replace π 2 = ind H H(C n ,W (Dn),k 1 ) (π) by π 2 = ind H H(C n ,W (Dn),k 1 ) (π). The map wf is a linear bijection H → H, so Res W (Bn) π 2 = Res W (Bn) π 2 . (i') We claim that in case (i) π 2 is still irreducible. As vector spaces V π 2 = V π ⊕ s en V π . For any nonzero linear subspace V of V π 2 , the irreducibility of π 2 tells us that there exists an h ∈ H such that π 2 (h)V ⊂ V . For k 2 > 0 very small, the corresponding element of H still satisfies π 2 (h)V ⊂ V . This verifies the claim (i').
(ii') In case (ii), we claim that π 2 is reducible. From Theorem 6.1 we know that (6.12) |Irr 0 (H)| = |Irr(W (B n ))| = |Irr 0 (H )| (i) and (ii) provide a way to count the right hand side: • every s en -orbit of length two in Irr 0 (H(C n , W (D n ), k 1 )) contributes one, • every s * en -fixed element of Irr 0 (H(C n , W (D n ), k 1 )) contributes two The restriction of any element of Irr 0 (H) to H(C n , W (D n ), k 1 ) has all irreducible constituents in Irr 0 (H(C n , W (D n ), k 1 )). By Frobenius reciprocity, this implies that it is a constituent of π 2 for some π ∈ Irr 0 (H(C n , W (D n ), k 1 )).
When s * en (π) ∼ = π, we saw in (i') that {π, s * en (π)} contributes just one representation to Irr 0 (H). In case s * en (π) ∼ = π, π 2 can be reducible. It has length at most two, because that is its length as H(C n , W (D n ), k 1 )-module. When π 2 would contribute only one representation to Irr 0 (H), the sum of the contributions from the cases (i') and (ii') would be strictly smaller than the corresponding sum of the contributions from (i) and (ii) to Irr(H ). However, that would contradict (6.12). We conclude that (ii') holds.
A π 2 as in (ii') has length 2, and both its irreducible constituents become isomorphic to π upon restriction to H(C n , W (D n ), k 1 ). As Res W (Bn) (π 2 ) = ind W (Bn) W (Dn) Res W (Dn) (π), the restriction to W (B n ) of the two constituents of π 2 must be Res W (Dn) (π) ⊗ triv and Res W (Dn) (π) ⊗ sign. Here Res W (Dn) (π) extends to a representation of W (B n ), while triv and sign are representations of W (B n )/W (D n ). In combination with case (i') we see that Res W (Bn) (Irr 0 (H )) = Res W (Bn) (Irr 0 (H)).
Hence there is a unique bijection Irr 0 (H) → Irr 0 (H ) that preserves W -types.
Finally, we settle the remaining cases of Theorem 6.2.
Proof. Lemma 6.6 establishes Theorem 6.2 for the algebras in (6.11). Moreover (6.9) gives us a function f R,k that defines a useful partial order on Irr 0 (H) and Irr (W ). With Lemma 6.7 we transfer all that to H(C n , W (B n ), k) with k 2 /k 1 ∈ (0, 1/2). Then we have Theorem 6.2 whenever R is irreducible and spans a * . As noted in the proof of Lemma 6.3, that implies Theorem 6.2 for all H (still with the condition on the parameters for type F 4 components). We finish the proof by applying Lemmas 6.5 and 6.6 another time.
Suppose that H = H(G, L, L, r)/(r−r) for a cuspidal local system L on a nilpotent orbit for L (as in Paragraph 5.2). In terms of Theorem 5.7, ζ H from Theorem 6.2 is just the map M y,0,r,ρ → M y,0,0,ρ . Here (y, ρ) → M y,0,0,ρ is the generalized Springer correspondence from [Lus1], twisted by the sign character of W . So, for a graded Hecke algebra that can be constructed with equivariant homology, Theorem 6.2 recovers a generalized Springer correspondence for W .
Let us relax this notion, and call any nice parametrization of Irr(W ) a generalized Springer correspondence. Then Theorem 6.2 qualifies as such, and we can regard ζ H : Irr 0 (H) → Irr(W ) as a "generalized Springer correspondence with graded Hecke algebras". This point of view has been pursued in [Slo1], where Irr 0 (H(C n , W (B n ), k)) has been parametrized with combinatorial data that mimic the above pairs (y, ρ).

A generalized Springer correspondence with affine Hecke algebras.
With Corollary 3.30 we can translate Theorem 6.2 into a statement about all tempered irreducible representations of H = H(R, λ, λ * , q), in relation with tempered C[X W ]-representations. We want to generalize that to all irreducible Hrepresentations, at least when q s ≥ 1 for all s ∈ S aff .
With Corollary 3.30 we will transfer Proposition 6.10 to affine Hecke algebras with parameters in R ≥1 . However, there is one detail that we need to get out of the way first. Let λ, λ * : R → R ≥0 be parameter functions. Then q sα = q λ(α) and q λ * (α) s α belong to R ≥1 for all α ∈ R, but λ(α) − λ * (α) could be < 0. In that case Theorem 3.29 could produce a graded Hecke algebra with a negative parameter k u (α), to which we could not apply Proposition 6.10.
Proof. By Lemma 6.11 and the asserted naturality of ζ H and ζ H , we may assume that λ(α) ≥ λ * (α) for all α ∈ R. For any u ∈ T un , the parameter function k u from (3.18) takes values in R ≥0 .
For any irreducible component R i of R not of type F 4 , we claim that R i does not possess a root subsystem isomorphic to F 4 . This can be seen with a case-by-case consideration of irreducible root systems. To get a root subsystem of type F 4 , R i needs to possess roots of different lengths, so it has type B n , C n , F 4 or G 2 . The rank of G 2 is too low, so R ∼ = G 2 . In type B n (resp. C n ) the short (resp. long) roots form a subsystem of type (A 1 ) n , and that does not contain D 4 . As both the long and the short roots in F 4 form a root system of type D 4 , we can conclude that B n ∼ = R i ∼ = C n . Now we apply Corollary 3.30 and reduce the theorem to the graded Hecke algebras H(t, W (R u ), k u ) Γ u , for all u ∈ T un . By the above, the parameter function k u fulfills the requirements of Theorem 6.2. Finally, we apply Proposition 6.10.
We note that Theorem 6.12 provides a natural bijection between the irreducible or standard modules of two affine Hecke algebras with the same root datum but different parameters q s ≥ 1.
When H arises from a cuspidal local system L on a nilpotent orbit for a Levi subgroup L of G, Theorem 6.12 boils down to Theorem 5.10. More precisely, in this case • the constructions that led to ζ H are analogous to those behind Theorem 5.10, • for graded Hecke algebras from cuspidal local systems we imposed such compatibility in the proof of Lemma 6.3. Theorem 5.10 with q = 1 can be considered as a generalized Springer correspondence for the (extended) affine Weyl group X Γ = W aff Ω, with geometric data (s, u, ρ). Consequently Theorem 6.12 can also be regarded as a generalized Springer correspondence of sorts, where the geometric data have been replaced by standard or irreducible modules of an affine Hecke algebra with (nearly arbitrary) parameters q s ∈ R ≥1 .