Periodic cyclic homology of affine Hecke algebras

This is the author’s PhD-thesis, which was written in 2006. The version posted here is identical to the printed one. Instead of an abstract, the short list of contents: Preface 5 1 Introduction 9 2 K-theory and cyclic type homology theories 13 3 Affine Hecke algebras 61 4 Reductive p-adic groups 103 5 Parameter deformations in affine Hecke algebras 129 6 Examples and calculations 169 A Crossed products 223 Bibliography 227 Index 237 Samenvatting 245 Curriculum vitae 253

💡 Research Summary

The dissertation “Periodic cyclic homology of affine Hecke algebras” presents a comprehensive study of the periodic cyclic homology (PCH) of affine Hecke algebras and situates the results within the broader contexts of K‑theory, representation theory of reductive p‑adic groups, and deformation theory. The work is organized into six substantive chapters plus an appendix on crossed products.

Chapter 1 introduces the motivation: affine Hecke algebras arise as the Iwahori–Hecke algebras of reductive p‑adic groups and thus provide a bridge between non‑commutative algebra and the harmonic analysis on p‑adic groups. Understanding their homological invariants promises insight into the Baum–Connes conjecture for these groups and into the Chern character linking K‑theory with cyclic homology.

Chapter 2 reviews the necessary background. It recalls algebraic K‑theory, Hochschild homology, cyclic homology, and Connes’ periodic cyclic homology. The Chern character is constructed for Banach algebras and shown to induce an isomorphism after tensoring with ℂ, provided the algebra satisfies suitable finiteness conditions. Spectral sequences and homological perturbation lemmas are introduced as technical tools for later calculations.

Chapter 3 develops the structure theory of affine Hecke algebras. Starting from a reduced root system Φ and a set of complex parameters qα, the algebra H(Φ,q) is defined via generators T_s (simple reflections) and relations that deform the group algebra of the affine Weyl group. The Bernstein decomposition splits H into a finite direct sum of “blocks” each Morita equivalent to a crossed product ℂ