Baum, P. F., Higson, N. and Plymen, R. J.
(2000)
Representation theory of p-adic groups: a view from operator algebras.
In,
*The Mathematical Legacy of Harish-Chandra. *
(Proceedings of Symposia in Pure Mathematics, 68)
American Mathematical Society, pp. 111-149.
(doi:10.1090/pspum/068/1767895).

## Abstract

Over the past several years, operator algebraists have become increasingly interested in the problem of calculating the K-theory of group C∗-algebras. The focal point of research in this area ist he BaumConnesConjecture[BCH],which proposes a description of K-theory for the C∗-algebra of a group in terms of homology and the representation theory of compact subgroups. Although the main applications of the Baum-Connes Conjecture are to issues in geometry and topology, the conjecture also appears to be of interest from the point of view of harmonic analysis. Whereas for applications to topology one is concerned with discrete groups G (arising as the fundamental groups of manifolds), the conjecture’s links with harmonic analysis appear to be the strongest for reductive Lie groups and p-adic groups. The purpose of these notes is to convey to a reasonably broad audience some byproducts of the authors’ research into the C∗-algebra K-theory of the p-adic group GL(N), which culminated in a proof of the Baum-Connes Conjecture in this case [BHP2]. Along the way to the proof a number of interesting issues came to light which we feel deserve some exposure, even though our understanding of them is far from complete, and is indeed mostly very tentative. Much of what follows is focused on what we call here chamber homology, which is a type of equivariant homology associated to the action of a reductive p-adic group on its Bruhat-Tits aﬃne building. The problem of computing chamber homology can be approached from a number of diﬀerent directions. An especially interesting problem is to reconcile chamber homology with the Bernstein decomposition for representations of reductive p-adic groups [Be, BD]. This appears to be a far from trivial matter, even in comparatively simple cases. In Section 5 we formulate two very general conjectures which give a broad description of a Bernstein decomposition in chamber homology (perhaps we should call our conjectures questions, since the evidence we have gathered in their favor is not overwhelming).

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