On the Spectral Decomposition of Affine Hecke Algebras

Abstract

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.

On calibrated representations and the Plancherel Theorem for affine Hecke algebras

“…The Plancherel Theorem has been proven in general by Heckman and Opdam [10] and Opdam [19] in a veritable tour-de-force parallelling Harish-Chandra's work [9] on the Plancherel Theorem for real and p-adic Lie groups (see also Reeder [26]). The Plancherel Theorem has been further developed by Delorme-Opdam, Opdam, Opdam-Solleveld, and Ciubotaru-Kato-Kato (see [4,6,20,21]).…”

“…Firstly, while the general formulation of the Plancherel Theorem in [19] is essentially complete, there are some constants that are not explicitly computed (they are conjectured in [19,Conjecture 2.27] to be rational numbers). Thus it is desirable to have a complete and direct calculation in ranks 1 and 2 which evaluate all constants involved.…”

“…The starting point and general philosophy for our derivation of the Plancherel Theorems is similar to that in [19], but since we restrict to the rank 1 and 2 cases the calculations can be carried out by hand. In fact, our calculations form an extension of Matsumoto's influential rank 1 calculations [16, §2.6], and hence provides a companion piece to [16] (see also Kutzko and Morris [13]).…”

“…In this paper, we extend Ram's combinatorial construction of calibrated representations of affine Hecke algebras to the multi-parameter case (including the non-reduced case), and we use these representations to derive explicit Plancherel formulae for all rank 1 and rank 2 affine Hecke algebras, following the work of Opdam (see [18,19]). …”

Conjectures about 𝑝-adic groups and their noncommutative geometry

“…The representations of a ne Hecke algebras have been subjected to a lot of study, see in particular [Lus4,KaLu,Opd,Sol2]. As a result the representation theory of H(T, R, q) is understood quite well, and close relations between Irr(H(T, R, q)) and Irr(H(T, R, 1)) ⇠ = (T //W (R)) 2 are known.…”