One of the far-reaching problems about continuous group actions is the w x BaumαConnes conjecture BCH . Although it is an assertion about certain K-theoretic invariants in the equivariant setting the understanding of the Ε½ . corresponding equivariant co homological invariants should be most useful. The literature contains various proposals how to construct in various situations, mostly for discrete groups or real Lie groups acting on mani-Ε½ . folds, the delocalized equivariant co homology. Now that we have the w x beautiful work of Bernstein and Lunts BL on the equivariant derived category of a real Lie group action this undoubtedly provides the correct framework in that case.
In this paper we consider the case of a locally compact and totally disconnected group G acting continuously on a locally compact space X. We develop the basic homological algebra of G-equivariant sheaves on X including Verdier duality. It turns out that certain functors, e.g., the direct image, do not correspond to their nonequivariant counterparts under the forgetful map. That this was to be expected was pointed out to me by Bernstein. We give a derived functor definition of delocalized equivariant homology and show that it unifies and generalizes the corresponding w x w x concepts in BC , BCH . Finally we relate the delocalized equivariant homology of the point to the cyclic homology of the Hecke algebra of G. w x w x This generalizes the main result of HN , Sch .
I thank P. Baum for several enlightening discussions about the paper w x BC and J. Bernstein for his helpful comments.