Equivariant cyclic homology and equivariant differential forms

Abstract

Let G be a compact Lie group, and let M be a compact manifold on which G acts smoothly.In this paper, we give a description of the equivariant periodic cyclic homology HP^ (C°° (M)) of C°° (M) as the cohomology of global equivariant differential forms on M: these are sections of a sheaf over the group G, whose stalk at g (E G is the complex of equivariant differential forms on the fixed-point set M 8 , with action of the centralizer 0 s .By the isomorphism HP^ (C°° (M)) '= K^ (M) (g)R (G) R 00 (G) with equivariant K-theory [where R 00 (G) is the space of smooth functions on G invariant under the adjoint action], we also obtain a de Rham description of equivariant K-theory.Let G a compact Lie group, and let M be a compact manifold on which G acts smoothly.Let R°° (G) be the ring C°° (G)° of smooth conjugation invariant functions on the group G; it is an algebra over the representation ring R(G) of G, since R(G) maps into R°° (G) by the character map.Then there is an equivariant Chem character ch^: K^ (M) = K^ (C°° (M)) -HP^ (C°° (M)) from the equivariant K-theory of M to the periodic cyclic homology HP^ (C°° (M)) of the algebra C°° (M) of smooth functions on M.This map induces an isomorphism HP^ (C°° (M)) ^ K^ (M)0R(G) R 00 (G); furthermore, there are graded-commutative products on both HP^ (C°° (M)) and K^ (M) such that the Chem character map is a ring homomorphism.These results are due to Block [3] (although he works with a crossed product involving algebraic functions instead of smooth ones), and Brylinski [5].In this paper, we will study the equivariant cyclic homology of the algebra C°° (M) in terms of equivariant differential forms on M; this extends the description which Hochschild-Kostant-Rosenberg gave of the Hochschild homology of C°° (M) in terms of differential ( ! ) This paper is dedicated to the memory of Ellen Block.