Affine Hecke Algebra with Unequal Parameters

• Affine Hecke algebras with unequal parameters are deformations of extended affine Weyl group algebras that assign independent positive parameters to each simple reflection.
• Their representation theory is classified via a parabolic induction process and an affine Springer correspondence, yielding precise parametrizations of irreducible and tempered modules.
• Scaling isomorphisms and Schwartz completions ensure that K-theory and cyclic homology remain invariant under the deformation of parameters.

An affine Hecke algebra with unequal parameters is a noncommutative algebraic structure arising as a deformation of the group algebra of an extended affine Weyl group, where each generator associated to a simple reflection is assigned an independent positive parameter. This generalizes the equal-parameter case and has crucial applications in the harmonic analysis on $p$-adic groups, geometric representation theory, and the local Langlands program. The representation theory, K-theory, and homological invariants of affine Hecke algebras with unequal parameters exhibit a deep and computable interplay with the geometry and combinatorics of torus–Weyl group actions, and their module categories admit precise parametrizations even in the presence of rich parameter variability.

1. Structure and Definition

Let $R$ be a root datum, and $W^e = X \rtimes (W_0 \rtimes \Gamma)$ the extended affine Weyl group. The affine Hecke algebra $H(R,q)$ with parameters $q : S \rightarrow \mathbb{R}_{>0}$ (one for each simple reflection $s$) is defined by generators $N_s$ subject to

• Quadratic relations: $(N_s + 1)(N_s - q(s)) = 0$ for all $s$,
• Braid relations corresponding to the Coxeter structure.

When all parameters coincide, this reduces to the classical Iwahori–Hecke algebra. Allowing unequal parameters, as first studied by Lusztig and developed in (Solleveld, 2010), gives rise to a much richer structure incorporating the variable root lengths or geometric data. For instance, in the context of $p$-adic groups, the parameters can encode different root multiplicities or ramification properties.

The algebra $H(R,q)$ is a deformation of the group algebra $\mathbb{C}[W^e]$, recovering it at $q(s) = 1$ for all $s$.

2. Representation Theory and Classification

The classification of irreducible representations of $H(R,q)$ with unequal parameters is achieved via a parabolic, Langlands-type induction:

1. Parabolic induction: Any irreducible $H$-representation is realized as the unique irreducible quotient of a standard module parabolically induced from discrete series representations of subalgebras. Explicitly,

$$\pi(P, \sigma, t) = \operatorname{Ind}_{H_P}^H(\sigma \circ \varphi_t)$$

where $P \subset F_0$ (simple roots), $\sigma$ is a discrete series $H_P$-module, and $t$ is an induction parameter in a subtorus $T^P$.

1. Springer correspondence: There exists a canonical (though nonnatural) correspondence

$\zeta: \operatorname{Irr}(H(R,q)) \to \text{(Virtual reps of $W^e$)}$

which, modulo torsion, induces an isomorphism of Grothendieck groups:

$$\zeta^\vee: G(H(R,q)\rtimes\Gamma) \otimes \mathbb{C} \cong G(C^\infty(T_{un})\rtimes(W_0\rtimes\Gamma)) \otimes \mathbb{C}$$

The set of images under $\zeta$ forms a $\mathbb{Z}$-basis of the representation ring of $W^e$.

1. Tempered and discrete series: The full classification reduces to the classification for tempered modules and, ultimately, the discrete series, reflecting the structure of representations in the archimedean, $p$-adic, and Langlands dual settings.

3. Geometric Realization via Schwartz Completion and Scaling Maps

A fundamental innovation is the use of the Schwartz completion $S(R,q)$, the completion of $H(R,q)$ under certain seminorms:

$$S(R,q) = \left\{ h = \sum_{w} h_w N_w : \sup_w |h_w| (N(w)+1)^n < \infty \ \forall n \right\}$$

Tempered representations of $H(R,q)$ extend to $S(R,q)$.

To interpolate between unequal- and equal-parameter cases, one defines

$$q^\epsilon(w) = q(w)^\epsilon \quad (\epsilon \in [0,1])$$

giving a path of algebras $H(R, q^\epsilon)$; $q^0$ is the trivial parameter, yielding the group algebra.

The scaling isomorphisms

$\nabla_\epsilon : S(R, q^\epsilon) \to S(R, q)$

are algebra isomorphisms for $\epsilon > 0$, and as $\epsilon \to 0$ yield an injection

$\nabla_0 : S(W^e) = S(R, q^0) \to S(R, q)$

which, while not surjective, induces isomorphisms on Grothendieck groups and K-theory modulo torsion, establishing that the representation theory and topological invariants of $S(R,q)$ are insensitive to deformation parameters.

4. K-theory and Noncommutative Geometric Invariants

The paper proves a version of the Higson–Plymen conjecture: the K-theory (as well as periodic cyclic homology) of the $C^*$-completion of $H(R,q)$ is independent of the parameter function $q$ up to torsion. That is,

$$K_*(C^*(W)\rtimes\Gamma)\otimes\mathbb{C} \cong K_*(C^*(R,q)\rtimes\Gamma)\otimes\mathbb{C}$$

also for topological periodic cyclic homology:

$$HP_*(H(R,q)\rtimes \Gamma) \cong HP_*(C^*(R,q)\rtimes \Gamma)$$

This independence is established through the scaling isomorphisms $\nabla_\epsilon$ and analytic continuation, demonstrating that noncommutative geometric invariants of these algebras remain constant under deformation of the parameters.

5. Affine Springer Correspondence and Duality

The construction of the affine Springer correspondence in the unequal parameter setting yields a bijection (after extension to the complexified Grothendieck group) between finite-dimensional representations of $H(R,q)$ and those of $W^e$. On tempered modules, this is realized by restriction to the Weyl group part of the central character; in this sense, the noncommutative dual of $H(R,q)$ is "the same" as that of $W^e$.

Explicitly, for the tempered dual, one obtains that

$$G(S(R,q)\rtimes\Gamma) \cong G(S(W^e)\rtimes\Gamma)$$

and after Fourier transform and Chern character, this isomorphism extends to periodic cyclic homology and K-theory.

6. Moduli and Analytical Properties

The representation theory depends analytically, via scaling of parameters, on the deformation path $q^\epsilon$. Families of modules vary smoothly, and the structure of the tempered dual is organized by continuous bijections (modulo torsion) between the dual spaces of $H(R,q)$ and $W^e$. While the map on irreducibles does not always preserve irreducibility, the image forms a basis in the sense described above.

7. Summary Table of Key Correspondences

Structure Level	$q \equiv 1$ (undeformed)	$q$ arbitrary, positive, unequal
Algebra	$\mathbb{C}[W^e]$	$H(R,q)$
Schwartz completion	$S(W^e)$	$S(R, q)$
Tempered dual	Reps of $W^e$	Tempered modules for $H(R, q)$
Grothendieck group	$G(W^e)$	$G(H(R, q))$
K-theory ($C^*$-alg)	$K_*(C^*(W)\rtimes \Gamma)$	$K_*(C^*(R, q)\rtimes \Gamma)$
Periodic cyclic hom.	$HP_*(C^*(W)\rtimes \Gamma)$	$HP_*(C^*(R, q)\rtimes \Gamma)$

Isomorphisms among these objects, modulo torsion (and after complexification where indicated), are induced by the scaling isomorphisms and the affine Springer correspondence.

8. Implications and Applications

This framework unifies representation-theoretic and noncommutative geometric methods for affine Hecke algebras, demonstrating that the underlying structure is stable under parameter variation. Consequences include:

• Explicit classification of irreducible and tempered modules via induction data.
• Affine analogs of the classical Springer correspondence for representation rings and K-theory.
• proof that K-theoretic and cyclic-homology invariants are deformation-invariant, up to torsion.
• Petrification of noncommutative geometric duality between affine Hecke algebras and extended affine Weyl groups.
• Utility in the paper of local Langlands correspondences, harmonic analysis on $p$-adic groups, and the formalism of categorical geometric representation theory.

This synthesis highlights the central role of the affine Springer correspondence, scaling homomorphisms, and analytic completions in controlling the structure and invariants of affine Hecke algebras with unequal parameters (Solleveld, 2010).