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, with action of the centralizer 0. By the isomorphism HP^ (C°° (M)) '= K^ (M) (g)R (G) R (G) with equivariant K-theory [where R (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 (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 HochschildKostant-Rosenberg gave of the Hochschild homology of C°° (M) in terms of differential () This paper is dedicated to the memory of Ellen Block. ANNALES SCIENTIFIQUES DE L'ECOLE NORMALE SUPERIEURE. 0012-9593/94/047$ 4.00/© Gauthier-Villars 494 J. BLOCK AND E. GETZLER forms on M, which was extended by Connes to cyclic homology. Let us give a rough idea of how this works. If c= /o^—^/ fc^ fi^C°°(M) and ^ G C°° (G), we define a map from the Lie algebra 0 of G to the space of A:-forms f^(M) on M, by the formula X^(expX) / fod(e-.f,)/\'-/\d{e-.fk)dt^dtk. J^k Here, A/: is the ^-simplex { ( ^ i , . . . , 4)|0 < ^i ^ t 2 . . . ^ tk < 1} c R^. This definition extends to define a map from C (M^ x G) to C (5, (M)), which moreover commutes with the actions of G on these two spaces: these actions are defined as follows: on C°° (M^ x G) by (h.c) ( x o , . . . , X k g ) = c(h~ xo,...,h~ Xk\h~ gh), and on C (5, (M)) by ( fa .o ; ) (X) = L ^ i C c ; ( a d ( f a ) X ) . Thus, we obtain a map from C^ (C°° (M)) = C (M^ x G)° to C (0, (M))° = (C[fl] 0 (M)) ^[s] C (0)°. This map is just one component of our equivariant Hochschild-Kostant-Rosenberg map; the other components correspond to other points of G, and define maps from C^ (C°° (M)) to C (^, (M))', where M is the fixed-point set of g acting on M, G is the fixedpoint set of g acting by conjugation on G (in other words the centralizer of g) and Q is the Lie algebra of G. In the above notation, this map is induced by sending /o • • • fk to X e ^ ^ ^ ( ^ e x p X ) / /od(et l x .A)A••.Ad(et f c x .^) |M.^l•••d4. ^Afc We call this map Og. It turns out that the correct way to describe the situation is by means of sheaves on G, with the topology given by open sets invariant under conjugation; all of our sheaves will be equivariant. In Section 1, we define a sheaf whose stalk at g G G is the space of germs at 0 of maps from Q to f2* (M) invariant under the centralizer G. In Section 2, we introduce the equivariant cyclic chains; these are just smooth functions on M^ x G which are invariant under the action of G: c (xo,..., X k , g) = c (h~ XQ, . . . , h~ xj,, h~ gh) for all h G G. It is easy to see how to define the sheaf C, (C (M), G) of equivariant ^-chains over G: the space of sections C^ (C (M)) over the invariant open set U is the space of invariant smooth functions on M^ x U. 4' SERIE TOME 27 1994 N° 4 EQUIVARIANT CYCLIC HOMOLOGY 495 The maps {ag\g G G} assemble to define a map of sheaves a : C. (C°° (M), G) -^ (M, G). The main result of this paper is the following equivariant generalization of the theorems of Hochschild-Kostant-Rosenberg and Connes; in a sense, we are completing the program of Baum-Brylinski-MacPherson. THEOREM. The map a defines a quasi-isomorphism of complexes of sheaves a: (C.(C°°(M), G), & + z t B ) ^ ( ^ ( M , G), i+ud). Taking the homology of both sides, we see that HP^ (C (M)) ^ IT (^ (M), d + .), where A^ (M) = F(G, O* (M, G)) is the space of global equivariant differential forms. In combination with the result relating equivariant K-theory with equivariant periodic cyclic homology, we obtain the following theorem: K^ (M) 0R (G) R°° (G) ^ K (^ (M), d + .). This work is heavily influenced by the papers of Baum-Brylinski-MacPherson [I], Berline-Vergne [2], and Brylinski [4]. We would like to thank M. Vergne and the referee for a number of helpful suggestions. The paper was written while the first author was at MIT and at the Courant Institute. The second author would like to thank the MSRI and the ENS for their hospitality during the writing of parts of this paper. Both authors are partially funded by the NSF.