[Paper Review] On the classification of irreducible representations of affine Hecke algebras with unequal parameters

Maarten Solleveld|arXiv (Cornell University)|Aug 1, 2010

Advanced Algebra and Geometry48 references4 citations

TL;DR

This paper establishes a geometric correspondence between the irreducible representations of affine Hecke algebras with unequal parameters and those of their associated Weyl groups, proving a conjecture of Aubert, Baum, and Plymen by showing that the Grothendieck groups of finite-dimensional representations are isomorphic modulo torsion. It further constructs continuous parameter deformations of the Schwartz completion, demonstrating that K-theory and periodic cyclic homology are invariant under parameter scaling, thus confirming a conjecture of Higson and Plymen on the parameter independence of C*-algebra K-theory.

ABSTRACT

Let $R$ be a root datum with affine Weyl group $W^e$, and let $H = H (R,q)$ be an affine Hecke algebra with positive, possibly unequal, parameters $q$. Then $H$ is a deformation of the group algebra $\mathbb C [W^e]$, so it is natural to compare the representation theory of $H$ and of $W^e$. We define a map from irreducible $H$-representations to $W^e$-representations and we show that, when extended to the Grothendieck groups of finite dimensional representations, this map becomes an isomorphism, modulo torsion. The map can be adjusted to a (nonnatural) continuous bijection from the dual space of $H$ to that of $W^e$. We use this to prove the affine Hecke algebra version of a conjecture of Aubert, Baum and Plymen, which predicts a strong and explicit geometric similarity between the dual spaces of $H$ and $W^e$. An important role is played by the Schwartz completion $S = S (R,q)$ of $H$, an algebra whose representations are precisely the tempered $H$-representations. We construct isomorphisms $ζ_ε: S (R,q^ε) o S (R,q)$ $(ε>0)$ and injection $ζ_0 : S (W^e) = S (R,q^0) o S (R,q)$, depending continuously on $ε$. Although $ζ_0$ is not surjective, it behaves like an algebra isomorphism in many ways. Not only does $ζ_0$ extend to a bijection on Grothendieck groups of finite dimensional representations, it also induces isomorphisms on topological $K$-theory and on periodic cyclic homology (the first two modulo torsion). This proves a conjecture of Higson and Plymen, which says that the $K$-theory of the $C^*$-completion of an affine Hecke algebra $H (R,q)$ does not depend on the parameter(s) $q$.

Motivation & Objective

• To establish a natural correspondence between irreducible representations of affine Hecke algebras with unequal parameters and those of the associated Weyl group.
• To prove the Aubert–Baum–Plymen conjecture on the geometric similarity between the dual spaces of affine Hecke algebras and Weyl groups.
• To demonstrate that the K-theory and periodic cyclic homology of the C*-completion of affine Hecke algebras are independent of the parameter values, confirming a conjecture of Higson and Plymen.
• To construct continuous parameter deformations of the Schwartz algebra and analyze their behavior under scaling to the Weyl group case.
• To show that unitary and tempered representations behave continuously under parameter deformation, preserving key representation-theoretic invariants.

Proposed method

• Define a map from irreducible representations of the affine Hecke algebra H(R,q) to those of the Weyl group W, which extends to an isomorphism on Grothendieck groups modulo torsion.
• Construct the Schwartz completion S(R,q) of the affine Hecke algebra, whose representations are precisely the tempered ones.
• Define continuous scaling maps ζε: S(R,q^ε) → S(R,q) for ε > 0 and an injection ζ₀: S(W) → S(R,q), which behaves like an isomorphism in K-theory and cyclic homology.
• Use the theory of parabolic induction and R-groups to analyze reducibility and confluence of residual spectra under parameter deformation.
• Apply results from noncommutative geometry, including topological K-theory and periodic cyclic homology, to compare algebras across different parameter values.
• Leverage the Langlands classification and geometric parametrization of representations to analyze the structure of the tempered dual space.

Experimental results

Research questions

• RQ1 How do the irreducible representations of affine Hecke algebras with unequal parameters relate to those of the Weyl group?
• RQ2 To what extent is the tempered dual space of an affine Hecke algebra with unequal parameters geometrically similar to that of the Weyl group?
• RQ3 Does the K-theory of the C*-completion of an affine Hecke algebra depend on the parameter values?
• RQ4 How do unitary and tempered representations deform continuously as parameters are scaled to 1?
• RQ5 What happens to the reducibility of parabolically induced representations at confluence points of residual spectra?

Key findings

• The map from irreducible H-representations to W-representations induces an isomorphism on Grothendieck groups of finite-dimensional representations modulo torsion.
• The continuous scaling maps ζε: S(R,q^ε) → S(R,q) exist and are isomorphisms for ε > 0, with ζ₀: S(W) → S(R,q) inducing isomorphisms on topological K-theory and periodic cyclic homology modulo torsion.
• The Aubert–Baum–Plymen conjecture is confirmed: the dual spaces of H(R,q) and W are geometrically similar, with confluences of residual points matching across parameter deformations.
• For small parameter perturbations, the number of tempered irreducibles with central character in a fixed region U/W₀ remains stable, though W₀-types may change.
• At critical parameter values (e.g., q₀ = q₁ = q₂), certain parabolically induced representations become reducible, with two irreducible components, as verified via R-group analysis and the Deligne–Langlands–Kazhdan–Lusztig parametrization.
• The K-theory and periodic cyclic homology of the C*-completion of H(R,q) are independent of the parameter q, confirming the Higson–Plymen conjecture.