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Representation theory of $$p$$-adic groups: A view from operator algebras. (English) Zbl 0982.19006
Doran, Robert S. (ed.) et al., The mathematical legacy of Harish-Chandra. A celebration of representation theory and harmonic analysis. Proceedings of an AMS special session honoring the memory of Harish-Chandra, Baltimore, MD, USA, January 9-10, 1998. Providence, RI: American Mathematical Society (AMS). Proc. Symp. Pure Math. 68, 111-149 (2000).
We cite from the introduction: “These notes present to a general audience some issues which arose during the authors’ study of $$C^*$$-algebra $$K$$-theory for the $$p$$-adic group $$GL(N)$$ – a study which culminated in a proof of the Baum-Connes conjecture for this group. We shall formulate a number of conjectures and questions which connect the Bernstein decomposition of smooth representations with a kind of equivariant homology for the affine building of a reductive group. For $$GL(N)$$ these issues are very closely related to the Bushnell-Kutzko theory of types.”
The survey paper under review is concerned with a rather detailed analysis of the cyclic homology of the Hecke algebra $${\mathcal H}(G)$$ of a $$p$$-adic reductive group $$G$$ and in particular its Bernstein decomposition. It begins with the definition of the chamber homology of the affine building, which incorporates both combinatorial and representation theoretic data of the family of compact subgroups of $$G$$. It is calculated in some examples and is shown to be closely related to the periodic cyclic homology of the Hecke algebra. A large portion of the paper is devoted to the study of the decomposition of chamber homology induced by the Bernstein decomposition of the variety of supercuspidal pairs of $$G$$. Several conjectures about this decomposition are made. Relating this to the theory of types of Bushnell and Kutzko the authors finally show that in the case $$G=GL(N)$$ the inclusion of the Hecke algebra $${\mathcal H}(G)$$ into the Schwartz algebra $${\mathcal S}(G)$$ induces an isomorphism on periodic cyclic homology. The fact that the latter homomorphism is an isomorphism can be interpreted as the content of a cyclic homology version of the Baum-Connes conjecture for $$p$$-adic reductive groups.
For the entire collection see [Zbl 0943.00049].

MSC:
 19K99 $$K$$-theory and operator algebras 46L80 $$K$$-theory and operator algebras (including cyclic theory)