We study the category O of representations of the rational Cherednik algebra A W attached to a complex reflection group W . We construct an exact functor, called Knizhnik-Zamolodchikov functor: O → H W -mod, where H W is the (finite) Iwahori-Hecke algebra associated to W . We prove that the Knizhnik-Zamolodchikov functor induces an equivalence between O/O tor , the quotient of O by the subcategory of A W -modules supported on the discriminant, and the category of finite-dimensional H W -modules. The standard A W -modules go, under this equivalence, to certain modules arising in Kazhdan-Lusztig theory of "cells", provided W is a Weyl group and the Hecke algebra H W has equal parameters. We prove that the category O is equivalent to the module category over a finite dimensional algebra, a generalized "q-Schur algebra" associated to W .

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An affine Hecke algebra H contains a large abelian subalgebra A spanned by the Bernstein-Zelevinski-Lusztig basis elements θx, where x runs over (an extension of) the root lattice. The centre Z of H is the subalgebra of Weyl group invariant elements in A. The natural trace ('evaluation at the identity') of the affine Hecke algebra can be written as integral of a certain rational n-form (with values in the linear dual of H) over a cycle in the algebraic torus T = Spec(A). This cycle is homologous to a union of 'local cycles'. We show that this gives rise to a decomposition of the trace as an integral of positive local traces against an explicit probability measure on the spectrum W 0 \ T of Z. From this result we derive the Plancherel formula of the affine Hecke algebra. ContentsOn the spectral decomposition of affine Hecke algebras(2.13) The Iwahori-Hecke algebra as a Hilbert algebraMany of the results of this subsection are well known (see [31]). Let R be a root datum, and let q be a real number with q > 1. We assume that for all s ∈ S aff we are given a real number f s . Throughout this paper we use the convention that the labels as discussed in the previous subsection are defined by Convention 2.1. The labels are of the form(2.14)We write q := (q(s)) s∈S aff for the corresponding label function on S aff . The following theorem is well known.Theorem 2.2. There exists a unique complex associative algebra H = H(R, q) with C-basis (T w ) w∈W which satisfy the following relations.(a) If l(ww ) = l(w) + l(w ), then T w T w = T ww .(b) If s ∈ S aff , then (T s + 1)(T s − q(s)) = 0.Proof . This is elementary and well known. Use the Frobenius-Schur theorem for (iii), Dixmier's version of Schur's lemma for (iv) and Proposition 2.10. Corollary 2.12 (see also [31]). Restriction to H induces an injection of the setĈ into the spaceĤ of finite-dimensional irreducible * -representations of H.Consequently, the C * -algebra C is of finite type I. Proof .Because H ⊂ C is dense, it is clear that a representation of C is determined by its restriction to H and that (topological) irreducibility is preserved. Hence, by the previous corollary, all irreducible representations of C have finite dimension.We equip T and W 0 \ T with the analytic topology. Given π ∈Ĉ, we denote by W 0 t π ∈ W 0 \ T the character of Z such that χ π (z) = dim(π)z(t π ) (note that π(Z) can not vanish identically since 1 ∈ Z).By Proposition 2.9, the * -operator on Z is such that z * (t) = z(t −1 ). When π ∈Ĉ, we have χ π (x * ) = χ π (x). It follows that t −1 π ∈ W 0 t π for all π ∈Ĉ. Let us denote byProposition 2.13. The map p z :Ĉ → W 0 \ T defined by p z (π) = W 0 t π is continuous and finite. Its image S = p z (Ĉ) ⊂ W 0 \ T Herm is the spectrumẐ of the closureZ of Z in C. The map p z :Ĉ → S is closed.Proof . It is clear that the image is in W 0 \ T Herm and that the map is finite (by Proposition 2.10). Since W 0 \ T is Hausdorff andĈ is compact, the map p z is closed if it is continuous.

SummaryThe graded Hecke algebra has a simple realization as a certain algebra of operators acting on a space of smooth functions. This operator algebra arises from the study of the root system analogue of Yang's system of n particles on the real line with delta function potential. It turns out that the spectral problem for this generalization of Yang's system is related to the problem of finding the spherical tempered representations of the graded Hecke algebra. This observation turns out to be very useful for both these problems. Application of our technique to affine Hecke algebras yields a simple formula for the formal degree of the generic Iwahori spherical discrete series representations.

We introduce the generic central character of an irreducible discrete series representation of an affine Hecke algebra. Using this invariant we give a new classification of the irreducible discrete series characters for all abstract affine Hecke algebras (except for the types E(1) 6,7,8 ) with arbitrary positive parameters and we prove an explicit product formula for their formal degrees (in all cases).