[Paper Review] Periodic cyclic homology of affine Hecke algebras

Maarten Solleveld|ArXiv.org|Oct 8, 2009

Advanced Algebra and Geometry83 references18 citations

TL;DR

This paper computes the periodic cyclic homology of affine Hecke algebras, a class of algebras arising in the representation theory of reductive $p$-adic groups. Using techniques from $K$-theory, cyclic homology, and harmonic analysis, it establishes a Chern character isomorphism that relates the periodic cyclic homology to the representation-theoretic data of the algebra, particularly in the case of equal parameters.

ABSTRACT

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

Motivation & Objective

• To compute the periodic cyclic homology of affine Hecke algebras, which are deformations of group algebras of affine Weyl groups.
• To establish a Chern character isomorphism between the periodic cyclic homology and the $K$-theory of the algebra.
• To extend known results on the representation theory of affine Hecke algebras—particularly in the equal parameter case—using homological invariants.
• To connect the homological structure of affine Hecke algebras to harmonic analysis on $p$-adic groups via the Bernstein decomposition.
• To provide a homological framework for understanding the classification of irreducible representations in terms of geometric and algebraic data.

Proposed method

• Applies topological cyclic theory and periodic cyclic homology to finite-type algebras, particularly affine Hecke algebras.
• Uses the Chern character to relate topological $K$-theory to periodic cyclic homology, leveraging the isomorphism in the equal parameter case.
• Employs the Fourier transform on the Hecke algebra to relate representations to functions on the dual space.
• Utilizes the Bernstein decomposition of the category of smooth representations into blocks, each equivalent to a module category over an affine Hecke algebra.
• Applies results from equivariant $K$-theory and the work of Kazhdan and Lusztig to classify irreducible representations and relate them to the homology computation.
• Relies on the structure of reductive $p$-adic groups and their Harish-Chandra Schwartz algebras to define the relevant algebras and their homological invariants.

Experimental results

Research questions

• RQ1 What is the periodic cyclic homology of an affine Hecke algebra, particularly in the case where all parameters are equal?
• RQ2 How does the periodic cyclic homology relate to the $K$-theory and representation theory of the algebra?
• RQ3 Can the Chern character provide an isomorphism between the periodic cyclic homology and the $K$-theory of the affine Hecke algebra?
• RQ4 How does the homological structure of the algebra reflect the Bernstein decomposition and Langlands classification of representations?
• RQ5 What role do tempered and discrete series representations play in the computation of periodic cyclic homology?

Key findings

• For affine Hecke algebras with equal parameters, the periodic cyclic homology is isomorphic to the $K$-theory of the algebra via the Chern character.
• The computation relies on the fact that in the equal parameter case, the irreducible representations of the affine Hecke algebra are in bijection with those of the underlying affine Weyl group, as established by Kazhdan and Lusztig.
• The Chern character induces an isomorphism between the periodic cyclic homology and the $K$-theory of the algebra, providing a homological realization of the representation-theoretic classification.
• The method extends to general positive parameters, though the classification is only partial, and the homology computation is more involved.
• The results show that the periodic cyclic homology captures essential representation-theoretic data, such as the structure of discrete series and tempered representations.
• The framework allows for the interpretation of the Langlands classification in terms of homological invariants, linking noncommutative geometry with harmonic analysis on $p$-adic groups.