K-theoretic deformation conjecture for extended affine Hecke algebras

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Let R\mathcal R be a root datum with affine Weyl group WeW^e, let Γ\Gamma be the finite group acting in the extended affine Hecke algebra, and let qq be a positive parameter function. Write C∗(We)⋊ΓC^*(W^e)\rtimes\Gamma for the group C∗C^*-algebra crossed product and C∗(R,q)⋊ΓC^*(\mathcal R,q)\rtimes\Gamma for the corresponding affine-Hecke C∗C^*-completion crossed product. K-theoretic deformation conjecture. The abelian groups

K∗(C∗(We)⋊Γ)andK∗(C∗(R,q)⋊Γ)K_*(C^*(W^e)\rtimes\Gamma)\quad\text{and}\quad K_*(C^*(\mathcal R,q)\rtimes\Gamma)

are naturally isomorphic. This is presented as the expected consequence of homotopy invariance of KK-theory under deformation of the parameter function; the source does not provide a separate resolution status for this formulation.

References

Primary source

Maarten Solleveld, “On the classification of irreducible representations of affine Hecke algebras with unequal parameters”, arXiv:1008.0177 (2013).

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