Equivariant Algebraic Index Theorem

EQUIVARIANT ALGEBRAIC INDEX THEOREM arXiv:1701.04041v1 [math.KT] 15 Jan 2017 ALEXANDER GOROKHOVSKY, NIEK DE KLEIJN, RYSZARD NEST Abstract. We prove a Γ-equivariant version of the algebraic index theorem, where Γ is a discrete group of automorphisms of a formal deformation of a symplectic manifold. The particular cases of this result are the algebraic version of the transversal index theorem related to the theorem of A. Connes and H. Moscovici for hypoelliptic operators and the index theorem for the extension of the algebra of pseudodifferential operators by a group of diffeomorphisms of the underlying manifold due to A. Savin, B. Sternin, E. Schrohe and D. Perrot. Contents 1. Introduction 1.1. The main result. 1.2. Structure of the article 2. Algebraic Index Theorem 2.1. Deformed formal geometry 2.2. Fedosov connection and Gelfand-Fuks construction 2.3. Algebraic index theorem in Lie algebra cohomology 2.4. Algebraic index theorem 3. Equivariant Gelfand-Fuks map
4. Pairing with HCper A~c ⋊ Γ • 5. Evaluation of the equivariant classes Appendix A. Cyclic/simplicial structure A.1. Cyclic homologies A.2. Replacements for cyclic complexes A.3. Lie algebra cohomology References 2 4 5 6 6 9 11 13 14 19 21 23 24 26 31 31 Alexander Gorokhovsky was partially supported by an NSF grant. Niek de Kleijn and Ryszard Nest were supported by the Danish National Research Foundation through the Centre for Symmetry and Deformation (DNRF92). Niek de Kleijn was also partially supported by the IAP “Dygest” of the Belgian Science Policy. 1 2 ALEXANDER GOROKHOVSKY, NIEK DE KLEIJN, RYSZARD NEST 1. Introduction The term index theorems is usually used to describe the equality of, on one hand, analytic invariants of certain operators on smooth manifolds and, on the other hand, topological/geometric invariants associated to their “symbols”. A convenient way of thinking about this kind of results is as follows. One starts with a C ∗ -algebra of operators A associated to some geometric situation and a K-homology cycle (A, π, H, D), where π : A → B(H) is a ∗-representation of A on a Hilbert space H and D is a Fredholm operator on H commuting with the image of π modulo compact operators K. The explicit choice of the operator D typically has some geometric/analytic flavour, and, depending on the parity of the K-homology class, H can have a Z/2Z grading such that π is even and D is odd. Given such a (say even) cycle, an index of a reduction of D by an idempotent in A ⊗ K defines a pairing of K-homology and K-theory, i. e. the group homomorphism K0 (C, A) × K0 (A, C) −→ Z. (1) One can think of this as a Chern character of D, ch(D) : K∗ (A) −→ Z, and the goal is to compute it explicitly in terms of some topological data extracted from the construction of D. Some examples are as follows. A = C(X), where X is a compact manifold and D is an elliptic pseudodifferential operator acting between spaces of smooth sections of a pair of vector bundles on X. The number < ch(D), [1] > is the Fredholm index of D, i. e. the integer Ind (D) = dim(Ker(D)) − dim(Coker(D)) and the Atiyah–Singer index theorem identifies it with the evaluation of the Â-genus of T ∗ X on the Chern character of the principal symbol of D. This is the situation analysed in the original papers of Atiyah and Singer, see [1]. A = C ∗ (F ), where F is a foliation of a smooth manifold and D is a transversally elliptic operator on X. Suppose that a K0 (A) class is represented by a projection p ∈ A, where A is a subalgebra of A closed under holomorphic functional calculus, so that the inclusion A ⊂ A induces an isomorphism on K-theory. For appropriately chosen A, the fact that D is transversally elliptic implies that the operator pDp is Fredholm on the range of p and the index theorem identifies the integer Ind (pDp) with a pairing of a certain cyclic cocycle on A with the Chern character of p in the cyclic periodic complex of A. For a special class of hypo-elliptic operators see f. ex. [6] Suppose again that X is a smooth manifold. The natural classP of representatives of Khomology classes of C(X) given by operators of the form D = γ∈Γ Pγ π(γ), where Γ is EQUIVARIANT ALGEBRAIC INDEX THEOREM 3 a discrete group acting on L2 (X) by Fourier integral operators of order zero and Pγ is a collection of pseudodifferential operators on X, all of them of the same (non-negative) order. The principal symbol σΓ (D) of such a D is an element of the C ∗ -algebra C(S ∗ X) ⋊max Γ, where S ∗ M is the cosphere bundle of M. Invertibility of σΓ (D) implies that D is Fredholm and the index theorem in this case would express Ind Γ (D) in terms of some equivariant cohomology classes of M and an appropriate equivariant Chern character of σΓ (D). For the case when Γ acts by diffeomorphisms of M, see [25, 19]. The typical computation proceeds via a reduction of the class of operators D under consideration to an algebra of (complete) symbols, which can be thought of as a ”formal deformation” A~ . Let us spend a few lines on a sketch of the construction of A~ in the case when the operators in question come from a finite linear combination of diffeomorphisms of a compact manifold X with coefficients in the algebra DX of differential operators on X. A special case is of course that of an elliptic differential operator on X. Example 1.1. Let Γ be a subgroup of the group of diffeomorphisms of X viewed as a • discrete group. Γ acts naturally on DX . Let DX be the filtration by degree of DX . Then the associated Rees algebra k R = {(a0 , a1 , . . .) | ak ∈ DX } with the product (a0 , a1 , . . .)(b0 , b1 , . . .) = (a0 b0 , a0 b1 + a1 b0 , . . . , X ai bj , . . .) i+j=k has the induced action of Γ. The shift ~ : (a0 , a1 , . . .) → (0, a0 , a1 , . . .) Q makes R into an CJ~K-module and R/~R is naturally isomorphic to k P olk (T ∗ X) where P olk (T ∗ X) is the space of smooth, fiberwise polynomial functions of degree k on the Q ∗ cotangent bundle T X. A choice of an isomorphism of R with k P olk (T ∗ X)J~K induces on Q ∗ ∞ ∗ k P olk (T X)J~K an associative, ~-bilinear product ⋆, easily seen to extend to C (T X). Since Γ acts by automorphisms on R, it also acts on (C ∞ (T ∗ X)J~K, ⋆). This is usually formalized in the following definition. Definition 1.2. A formal deformation quantization of a symplectic manifold (M, ω) is an associative CJ~K-linear product ⋆ on C ∞ (M)J~K of the form X i~ f ⋆ g = f g + {f, g} + ~k Pk (f, g); 2 k≥2 where {f, g} := ω(Iω (df ), Iω (dg)) is the canonical Poisson bracket induced by the symplectic structure, Iω is the isomorphism of T ∗ M and T M induced by ω, and the Pk denote bidifferential operators. We will also require that f ⋆ 1 = 1 ⋆ f = f for all f ∈ C ∞ (M)J~K. We will use A~ (M) to denote the algebra (C ∞ (M)J~K, ⋆). The ideal A~c (M) in A~ (M), 4 ALEXANDER GOROKHOVSKY, NIEK DE KLEIJN, RYSZARD NEST P consisting of power series of the form k ~k fk , where fk are compactly supported, has a unique (up to a normalization) trace T r with values in C[~−1 , ~K (see f. ex. [10]). It is not difficult to see that the index computations (as in 1) reduce to the computation of the pairing of the trace (or some other cyclic cocycle) with the K-theory of the symbol algebra, which, in the example above, is identified with a crossed product A~c (M) ⋊ Γ. An example of this reduction is given in [17]. Since the product in A~c (M) is local, the computation of the pairing of K-theory and cyclic cohomology of A~c (M) reduces to a differential-geometric problem and the result is usually called the “algebraic index theorem”. Remark 1.3. Since cyclic periodic homology is invariant under (pro)nilpotent extensions, the result of the pairing depends only on the ~ = 0 part of the K-theory of A~c (M) ⋊ Γ. In our example, the ~ = 0 part of the symbol algebra A~c (M) ⋊ Γ is just Cc∞ (M) ⋊ Γ, hence the Chern character of D enters into the final result only through a class in the equivariant cohomology HΓ∗ (M). 1.1. The main result. Suppose that Γ is a discrete group acting by continuous automorphisms on a formal deformation A~ (M) of a symplectic manifold M. Let A~ (M)⋊Γ denote the algebraic crossed product associated to the given action of Γ. For a non-homogeneous group cocycle ξ ∈ C k (Γ, C), the formula below defines a cyclic k-cocycle T rξ on A~c (M)⋊Γ. (2) T rξ (a0 γ0 ⊗ . . . ⊗ ak γk ) = δe,γ0 γ1 ...γk ξ(γ1 , . . . , γk )T r(a0 γ0 (a1 ) . . . (γ0 γ1 . . . γk−1 )(ak )). The action of Γ on A~ (M) induces (modulo ~) an action of Γ on M by symplectomorphisms. Let σ be the “principal symbol” map: A~ (M) → A~ (M)/~A~ (M) ≃ C ∞ (M). It induces a homomorphism σ : A~ (M) ⋊ Γ −→ C ∞ (M) ⋊ Γ, still denoted by σ. Let • Φ : HΓ• (M) −→ HCper (Cc∞ (M) ⋊ Γ) be the canonical map (first constructed by Connes) induced by (19), where HΓ• (M) denotes the cohomology of the Borel construction M ×Γ EΓ and Cc∞ (M) denotes the algebra of compactly supported smooth functions on M. The main result of this paper is the following.
~ Theorem 1.4. Let e, f ∈ MN A (M) ⋊ Γ be a couple of idempotents such that the
difference e − f ∈ MN A~c (M) ⋊ Γ is compactly supported, here A~c (M) denotes the ideal of compactly supported elements of A~ (M). Let [ξ] ∈ Hk (Γ, C) be a group cohomology class. Then [e] − [f ] is an element of K0 (A~c (M) ⋊ Γ) and its pairing with the cyclic cocycle T rξ is given by D   E (3) < T rξ , [e] − [f ] >= Φ ÂΓ eθΓ [ξ] , [σ(e)] − [σ(f )] . EQUIVARIANT ALGEBRAIC INDEX THEOREM 5 Here ÂΓ ∈ HΓ• (M) is the equivariant Â-genus of M (defined in section 5), θΓ ∈ HΓ• (M) is the equivariant characteristic class of the deformation A~ (M) (also defined in section 5). In the case when the action of Γ is free and proper, we recover the algebraic version of Connes-Moscovici higher index theorem. The above theorem gives an algebraic version of the results of [25], without the requirement that Γ acts by isometries. To recover the analytic version of the index theorem type results from [25] and[19] one can apply the methods of [17]. 1.2. Structure of the article. Section 2 contains preliminary material, extracted mainly from [3] and [7]. It is included for the convenience of the reader and contains the following material. • Deformation quantization of symplectic manifolds and Gelfand–Fuks construction. Following Fedosov, a deformation quantization of a symplectic manifold A~ (M) can be seen as the space of flat sections of a flat connection ∇F on the bundle W of Weyl algebras over M constructed from the bundle of symplectic vector spaces T ∗ M → M. The fiber of W is isomorphic to the Weyl algebra g = W (see definition 2.3) and ∇F is a connection with values in the Lie algebra of derivations of W, equivariant with respect to a maximal compact subgroup K of the structure group of T ∗ M. Suppose that L is a (g, K)-module. The Gelfand–Fuks construction provides a complex (Ω(M, L), ∇F ) of L-valued differential forms with a differential ∇F satisfying ∇2F = 0. Let us denote the corresponding spaces of cohomology classes by H ∗(M, L). An example is the Fedosov construction itself, in fact ( A~ (M) k = 0 k H (M, W) = 0 k 6= 0. The Gelfand–Fuks construction also provides a morphism of complexes ∗ GF : CLie (g, K; L)) −→ Ω(M, L) • Algebraic index theorem The Gelfand–Fuks map is used to reduce the algebraic index theorem for a deformation of M to its Lie algebra version involving only the (g, K)-modules given by the periodic cyclic complexes of W and the commutative algebra O = CJx1 , . . . , xn , ξ1 , . . . , ξn , ~K. In fact, the following holds. Theorem 1.5. Let L• = Hom−• (CC•per (W), Ω̂−• [~−1 , ~K[u−1 , uK[2d]). There exist two elements τ̂a and τ̂t in the hypercohomology group H0Lie (g, K; L• ) such that the following holds. (1) Suppose that M = T ∗ X for a smooth compact manifold X and A~ (M) is the deformation coming from the calculus of differential operators. Then whenever p and q are two idempotent pseudodifferential operators with p − q smoothing, Z M GF (τ̂a )(σ(p) − σ(q)) = T r(p − q) and Z M GF (τ̂t )(σ(p) − σ(q)) = Z M ch(p0 ) − ch(q0 ) 6 ALEXANDER GOROKHOVSKY, NIEK DE KLEIJN, RYSZARD NEST where T r is the standard trace on the trace class operators on L2 (M), p0 (resp. q0 ) are the ~ = 0 components of p (resp. q) and ch : K 0 (M) → H ev (M) is the classical Chern character. (2) h where Âf eθ̂ i i Xh θ̂ τ̂a = Âf e up τ̂t , p≥0 2p is the component of degree 2p of a certain hypercohomology class.   In the case of M = T ∗ X as above, GF Âf eθ̂ coincides with the Â-genus of M. 2p In general, given a group Γ acting on a deformation quantization algebra A~ (M), there does not exist any invariant Fedosov connection. As a result, the Gelfand–Fuks map described in section 2 does not extend to this case. The rest of the paper is devoted to the construction of a Gelfand–Fuks map that avoids this problem and the proof of the main theorem. Section 3 is devoted to a generalization of the Gelfand–Fuks construction to the equivariant case, where an analogue of the Fedosov construction and Gelfand–Fuks map are constructed on M × EΓ. Section 4 is devoted to a construction of a pairing of the periodic cyclic homology of the crossed product algebra with a certain Lie algebra cohomology appearing in Section 2. The main tool is for this construction is the Gelfand–Fuks maps from Section 3. Section 5 contains the proof of the main result. The appendix is used to define and prove certain statements about the various cohomology theories appearing in the main body of the paper. All the results and definitions in the appendices are well-known and standard and are included for the convenience of the reader. 2. Algebraic Index Theorem 2.1. Deformed formal geometry. Let us start in this section by recalling the adaptation of the framework of Gelfand-Kazhdan’s formal geometry to deformation quantization described in [16, 18] and [3]. For the rest of this section we fix a symplectic manifold (M, ω) of dimension 2d and its deformation quantization A~ (M). Notation 2.1. Let m ∈ M. (1) Jm∞ (M) denotes the space of ∞-jets at m ∈ M; Jm∞ (M) := lim C ∞ (M)/ (Im )k , ←− where Im is the ideal of smooth functions vanishing at m and k ∈ N. (2) Since the product in the algebra A~ (M) is local, it defines an associative, CJ~K∞ ~ (M) bilinear product ⋆m on Jm∞ (M). A\ m denotes the algebra (Jm (M)J~K, ⋆m ). Notation 2.2. EQUIVARIANT ALGEBRAIC INDEX THEOREM 7
~ (R2d ) , where the deformation A~ R2d has the prod(1) W will denote the algebra A\ 0 uct given by the Moyal-Weyl formula ! d i~ X (∂ξi ∂yi − ∂ηi ∂xi ) f (ξ, x) g (η, y) (f ⋆ g) (ξ, x) = exp . 2 i=1 ξ i =ηi xi =y i (2) Let x̂k , ξˆk denote the jets of xk , ξ k – the standard Darboux coordinates on R2d respectively. W has a graded algebra structure, where the degree of the x̂k ’s and ξˆk ’s is 1 and the degree of ~ is E D 2. (3) W will be endowed with the ~, x̂1 , . . . , x̂n , ξˆ1, . . . , ξˆn -adic topology. P k (4) We denote the (symbol) map given by ~ fk 7→ f0 by ~ (M) −→ J ∞ (M). σ̂m : A\ m m We shall also use the notation J0∞ (R2d ) =: O. Definition 2.3. For a real symplectic vector space (V, ω) we denote W(V ) := \ T (V ) ⊗R CJ~K . hv ⊗ w − w ⊗ v − i~ω(v, w)i \ Here T (V ) is the tensor algebra of V , T (V ) is its V -adic completion and the topology is given by the filtration by assigning elements of V degree 1 and ~ degree 2. The assignment V 7→ W(V ) is clearly functorial with respect to symplectomorphisms. Remark 2.4. Suppose (V, ω) is a 2d-dimensional real symplectic vector space. A choice of symplectic basis for V induces an isomorphism of CJ~K algebras: W(V ∗ ) ≃ W. b := Aut(W) denote the group of continuous CJ~K-linear automorNotation 2.5. Let G phisms of W. We let g = Der(W) denote the Lie algebra of continuous CJ~K-linear derivations of W. For future reference, let us state the following observation Lemma 2.6. The map 1 ad f ∈ g ~ Q is surjective. In particular, the grading of W induces a grading g = i≥−1 gi on g, namely the unique grading such that this map is of degree −2 (note that W0 = C is central) . We will use the notation Y g≥k = gi . W∋f → i≥k 8 ALEXANDER GOROKHOVSKY, NIEK DE KLEIJN, RYSZARD NEST Notation 2.7. Let g̃ = { 1~ f | f ∈ W} with a Lie algebra structure given by 1 1 1 [ f, g] = 2 [f, g] ~ ~ ~ and note that g̃ is a central extension of g. The corresponding short exact sequence has the form 1 ad (4) 0 −→ CJ~K −→ g̃ −→ g −→ 0, ~ where ad ~1 f (g) = ~1 [f, g]. The extension (4) splits over sp(2d, C) and, moreover, the corresponding inclusion sp(2d, C) ֒→ g̃ integrates to the action of Sp(2d, C). The Lie subalgebra sp(2d, R) ⊂ sp(2d, C) gets represented, in the standard basis, by elements of g̃ given by 1 1 k j −1 ˆk ˆj ξ ξ and x̂k ξˆj , where k, j = 1, 2, . . . , d. x̂ x̂ , ~ ~ ~ Lemma 2.8. The Lie algebra g≥0 has the structure of a semi-direct product g≥0 = g≥1 ⋊ sp(2d, C). b of automorphisms of W has a structure of a pro-finite dimensional Lie The group G b has the structure of group with the pro-finite dimensional Lie algebra g≥0 . As such, G semi-direct product b≃G b1 ⋊ Sp(2d, C), G b1 = exp g≥1 is pro-unipotent and contractible. The filtration of g induces a filtrawhere G b tion of G: b=G b0 ⊃ G b1 ⊃ G b2 ⊃ G b3 ⊃ . . . . G bk = exp(g≥k ). Here, for k ≥ 1, G Recall that A~ (M) is a formal deformation of a symplectic manifold (M, ω). In particular, ω gives an isomorphism T M → T ∗ M and the cotangent bundle T ∗ M will be given the induced structure of a symplectic vector bundle. For all m ∈ M, there exists a non-canonical isomorphism ~ (M) ≃ W. A\ m We collect them together in the following. Definition 2.9. n o ∼ \ ~ f M := ϕm | m ∈ M, ϕm : A (M)m −→ W b on M f endows it with the structure of G-principal b The natural action of G bundle. We give f the pro-finite dimensional manifold structure using the pro-nilpotent group structure M b of G/Sp(2d, C), see [16] for details. EQUIVARIANT ALGEBRAIC INDEX THEOREM 9 f is isomorphic to the trivial bundle M f× g Theorem 2.10. [16] The tangent bundle of M   f ⊗ g satisfying and there exists a trivialisation given by a g-valued one-form ω~ ∈ Ω1 M the Maurer-Cartan equation 1 dω~ + [ω~ , ω~ ] = 0. 2 For later use let us introduce a slight modification of the above construction. br = G b1 ⋊ Sp(2d, R). We will use M fr to denote the G br -principal Definition 2.11. Let G f consisting of the isomorphisms subbundle of M ∼ ~ (M) −→ ϕm : A\ W m such that ϕm , the reduction of ϕm modulo ~, is induced by a local symplectomorphism (R2d , 0) → (M, m). fr → M factorises through FM , the bundle of symplectic Note that the projection M br : frames in T M, equivariantly with respect to the action of Sp(2d, R) ⊂ G br G / Sp(2d, R) / fr M  FM  M. fr . We will use the same symbol for ω~ and its pull back to M b1 is con2.2. Fedosov connection and Gelfand-Fuks construction. Recall that G b1 -bundle M fr → FM admits a section F . Since tractible, thus in particular the principal G the space K is solid [26], we can choose F to be Sp(2d, R)-equivariant. Set AF = F ∗ ω~ ∈ Ω1 (FM ; g) . Since AF is Sp(2d, R)-equivariant and satisfies the Maurer-Cartan equation, (5) d + AF reduces to a flat g-valued connection ∇F on M, called the Fedosov connection. Example 2.12. Consider the case of M = R2d with the standard symplectic strucg 2d ture and let A~ (R2d ) denote the Moyal-Weyl deformation. Then both FR2d and R are trivial bundles. The trivialization is given by the standard (Darboux) coordinates x1 , . . . , xd , ξ 1 , . . . , ξ d. So we see, using the construction of ω~ in [16], that AF (X) = 1 [ω(X, −), −], where we consider ω(X, −) ∈ Γ(T ∗ M) ֒→ Γ(M; W). Let us denote the i~ generators of W corresponding to the standard coordinates by x̂i and ξˆi , then we see that AF (∂xi ) = −∂x̂i and AF (∂ξi ) = −∂ξ̂i . 10 ALEXANDER GOROKHOVSKY, NIEK DE KLEIJN, RYSZARD NEST Notation 2.13. Suppose that l ⊂ h is an inclusion of Lie algebras and suppose that the ad action of l on h integrates to an action of a Lie group L with Lie algebra l. An h module M is said to be an (h, L)-module if the action of l on M integrates to a compatible action of the Lie group L. If an (h, L)-module is equipped with a compatible grading and differential we will call it an (h, L)-cochain complex. Definition 2.14. We set n o Ω• (M; L) := η ∈ (Ω• (FM ) ⊗ L)Sp(2d) | ιX (η) = 0 ∀X ∈ sp(2d) for a (g, Sp(2d, R))-module L. Here the superscript refers to taking invariants for the diagonal action and ιX stands for contraction with the vertical vector fields tangent to the action of Sp(2n, R). Together with ∇F , (Ω• (M; L), ∇F ) forms a cochain complex. The same construction with a (g, Sp(2d, R)-cochain complex (L• , δ) yields the double complex (Ω• (M; L• ), ∇F , δ) Remark 2.15. Ω0 (M; L) is the space of sections of a bundle which we will denote by L, whose fibers are isomorphic to L. (Ω• (M; L), ∇F ) is the de Rham complex of differential forms with coefficients in L. Definition 2.16. Suppose that (L• , δ) is a (g, Sp(2d))-cochain complex. The Gelfand-Fuks • n map CLie (g, sp(2d); L• ) −→ Ω• (M; L• ) is defined as follows. Given ϕ ∈ CLie (g, sp(2d); L• ) and vector fields {Xi }i=1,...,n on FM set GF (ϕ)(X1 , . . . , Xn )(p) = ϕ(AF (X1 )(p), . . . , AF (Xn )(p)). Direct calculation using the fact that ω~ satisfies the Maurer-Cartan equation gives the following theorem: Theorem 2.17. The map GF is a morphism of double complexes • GF : (CLie (g, sp(2d); L• ), ∂Lie , δ) −→ (Ω• (M; L• ), ∇F , δ) . The change of Fedosov connection, i.e. of the section F , gives rise to a chain homotopic morphism of the total complexes. Example 2.18. (1) Suppose that L = C. The associated complex is just the de Rham complex of M. (2) Suppose that L = W. The associated bundle W(T ∗M), the Weyl bundle of M, is given by applying the functor W to the symplectic vector bundle T ∗ M. Moreover, the choice of F determines a canonical quasi-isomorphism JF∞ : (A~ , 0) −→ (Ω• (M; W), ∇F ) . (3) Suppose that L = (CC•per (W), b + uB), the cyclic periodic complex of W. The complex (Ω• (M; CC•per (W)), ∇F + b + uB) is a resolution of the jets at the diagonal of the cyclic periodic complex of A~ (M). Example 2.19. EQUIVARIANT ALGEBRAIC INDEX THEOREM 11 2 (1) Let θ̂ ∈ CLie (g, sp(2d); C) denote a representative of the class of the extension (3). ω The class of θ = GF (θ̂) belongs to i~ + H2 (M; C)J~K and classifies the deformations of M up to gauge equivalence (see e.g. [18]). (2) The action of sp(2d) on g is semisimple and sp(2d) admits a Sp(2d, R)-equivariant complement. Let Π be the implied Sp(2d, R)-equivariant projection g → sp(2d). Set R : g ∧ g −→ sp(2d) to be the two-cocycle R(X, Y ) = [Π(X), Π(Y )] − Π([X, Y ]). The Chern-Weil homomorphism is the map • CW : S • (sp(2d)∗ )Sp(2d) −→ H2Lie (g, sp(2d)) given on the level of cochains by CW (P )(X1, . . . , Xn ) = P (R(X1 , X2 ), . . . , R(Xn−1, Xn )). An example is the Â-power series Âf = CW det ad( X2 ) exp(ad( X2 )) − exp(ad(− X2 )) !! . GF (Âf ) = Â(T M), the Â-genus of the tangent bundle of M. 2.3. Algebraic index theorem in Lie algebra cohomology. Notation 2.20. We denote W(~) := W[~−1 ]. Notation 2.21. Our convention for shifts of complexes is as follows: (V • [k])p = V p+k . • ˆ Theorem 2.22 ([3],[4]).

Let (Ω̂ , d) denote the formal de Rham complex in 2d dimensions, and let C• W(~) , b denote the Hochschild complex of W(~) . (1) There exists a unique (up to homotopy) quasi-isomorphism  

µ~ : C•Hoch W(~) , b −→ Ω̂−• [~−1 , ~K[2d], d̂ . which maps the Hochschild 2d-chain   ϕ = 1 ⊗ Alt ξˆ1 ⊗ x̂1 ⊗ ξˆ2 ⊗ x̂2 ⊗ . . . ⊗ ξˆd ⊗ x̂d , P where Alt(z1 ⊗ . . . ⊗ zn ) := σ∈Σn (−1)sgnσ zσ(1) ⊗ . . . ⊗ zσ(n) , to the 0-form 1. µ~ extends to a quasi-isomorphism µ~ : (CC•per (W(~) ), b + uB) −→ (Ω̂−• [~−1 , ~K[u−1 , uK[2d], d̂). (2) The principal symbol map σ : W → W/~W ≃ O together with the HochschildKostant-Rosenberg map HKR given by 1 ˆ ˆ ˆ f0 ⊗ f1 ⊗ . . . ⊗ fn 7→ f0 df 1 ∧ df2 ∧ . . . ∧ dfn n! 12 ALEXANDER GOROKHOVSKY, NIEK DE KLEIJN, RYSZARD NEST induces a C-linear quasi-isomorphism   µ̂ : CC•per (W) −→ Ω̂• [u−1, uK, ud̂ . (3) The map of complexes J : (Ω̂• [u−1 , uK, ud̂) → (Ω̂−• [~−1 , ~K[u−1 , uK[2d], d̂) given by ˆ 1 ∧ . . . ∧ df ˆ n 7→ u−d−n f0 df ˆ 1 ∧ . . . ∧ df ˆ n. f0 df makes the following diagram commute up to homotopy,. (6) CC•per (W) ι / CC•per (W(~) ) µ~ / (Ω̂−• [~−1 , ~K[u−1 , uK[2d], d̂) . O σ J  CC•per (O) HKR /   Ω̂• [u−1 , uK, ud̂ Here the complex CC•per (W) at the leftmost top corner is that of W as an algebra over C: Remark 2.23. One can in fact extend the above C-linear ”principal symbol map” σ : CC•per (W) → CC•per (O) to a CJ~K-linear map of complexes CC•per (W) → CC•per (OJ~K), but we will not need it below. Notation 2.24. (1) Action of g by derivations on W extends to the complex CC•per (W) and we give it the corresponding (g, Sp(2d, R))-module structure. (2) The action of g on W taken modulo ~W, induces an action of g (by Hamiltonian vector fields) on (Ω̂−• , d) and hence on (Ω̂−• [~−1 , ~K[u−1 , uK[2d], d). We give (Ω̂−• [~−1 , ~K[u−1 , uK[2d], d) the induced structure of (g, Sp(2d, R))-module. (3) We set L• := Hom−• (CC•per (W), Ω̂−• [~−1 , ~K[u−1 , uK[2d]). L inherits the (g, Sp(2d, R))-module structure from the actions of g described above. The composition J ◦ HKR ◦ σ̂ is equivariant with respect to the action of g, hence the following definition makes sense.
Definition 2.25. τˆt is the cohomology class in the hypercohomology H0Lie g, sp(2d); L(~) given by the cochain (7) 0 J ◦ HKR ◦ σ ∈ CLie (g, Sp(2d, R); L0 ). Lemma 2.26. The cochain 0 µ~ ◦ ι ∈ CLie (g, Sp(2d, R); L0 ) extends to a cocycle in the complex • (CLie (g, Sp(2d, R); L• ), ∂Lie + ∂L ). EQUIVARIANT ALGEBRAIC INDEX THEOREM 13 The cohomology class of this cocycle is independent of
the choice of the extension. We will denote the corresponding class in H0Lie g, sp(2d); L(~) by τ̂a . For a proof of the next result see e.g. [3]. Theorem 2.27 (Lie Algebraic Index Theorem). We have i Xh τ̂a = Âf eθ̂ up τ̂t , 2p p≥0 i h θ̂ where Âf e 2p is the component of degree 2p of the cohomology class of Âf eθ̂ . 2.4. Algebraic index theorem. An example of an application of the above is the algebraic index theorem for a formal deformation of a symplectic manifold M. Note that we can view A~ as a complex concentrated in degree 0 and with trivial differential. Then, using the notation of remark 2.18, we find the quasi-isomorphism JF∞ : (A~ , 0) −→ (Ω• (M; W), ∇F ) . Similarly we find the quasi-isomorphism JF∞ : CC•per (A~ ) −→ Ω• (M; CC•per (W)). For future reference let us record the following observation. Lemma 2.28. The quasi-isomorphic inclusion C[~−1 , ~K[u−1 , uK ֒→ Ω̂−• [~−1 , ~K[u−1 , uK induces a quasi-isomorphism  
ι : Ω• (M)[~−1 , ~K[u−1, uK[2d], ddR −→ Ω• (M; Ω̂−• [~−1 , ~K[u−1, uK[2d]), ∇F + dˆ . Notation 2.29. We denote the inverse (up to homotopy) of ι by  
• −• −1 −1 ˆ T0 : Ω (M; Ω̂ [~ , ~K[u , uK[2d]), ∇F + d −→ Ω• (M)[~−1 , ~K[u−1 , uK[2d], ddR . For each Q ∈ Ω• (M; L• ) of total degree zero, we can define the map Q T 0 CQ : CC0per (A~ ) −→ Ω• (M; CC•per (W)) −→ Ω• (M; Ω̂−∗ [~−1 , ~K[u−1 , uK[2d]) −→ u−d R Ω•−∗ (M; C)[~−1 , ~K[u−1, uK[2d] −→M C[~−1 , ~K. Clearly CQ is a periodic cyclic cocycle if Q is a cocycle. We will apply this construction to the two cocycles τ̂t and τ̂a . Let us start with Cτ̂t . Tracing the definitions we get the following result. Proposition 2.30. Cτ̂t is given by n u w0 ⊗ . . . ⊗ w2n un−d 7 → (2n)! Z M σ(w0 )dσ(w1 ) ∧ . . . ∧ dσ(w2n ). 14 ALEXANDER GOROKHOVSKY, NIEK DE KLEIJN, RYSZARD NEST To get the corresponding result for Cτ̂a recall first that the algebra A~ (M) has a unique CJ~K-linear trace, up to a normalisation factor. This factor can be fixed as follows. Locally any deformation of a symplectic manifold is isomorphic to the Weyl deformation. Let U be such a coordinate chart and let ϕ : A~ (U) → A~ (R2d ) be an isomorphism. Then the trace T r is normalized by requiring that for any f ∈ A~c (U) we have Z ωd 1 ϕ(f ) . T r(f ) = (i~)d R2d d! Proposition 2.31. We have Cτ̂a = T r. Comments about the proof. First one checks that Cτ̂a is a 0-cocycle and therefore a trace. Hence it is a CJ~K-multiple of T r and it is sufficient to evaluate it on elements supported in a coordinate chart. Moreover, the fact that the Hochschild cohomology class of Cτ̂a is independent of the Fedosov connection implies that Cτ̂a is independent of it. Thus it is sufficient to verify the statement for R2d with the standard Fedosov connection. Let f ∈ A~c (R2d ), one checks that JF∞ (f ) ∈ Ω0 (R2d ; C0 (W)) is cohomologous to the element 1 2d • 2d ∞ f ϕ ω d ∈ Ω2d c (R ; C2d (W)) in Ωc (R ; C−• (W)) . It follows that the GF (τ̂a )JF (f ) is (i~)d d! cohomologous to (i~)1d d! f ϕ ω d (see the theorem 2.22) and therefore Z ωd 1 f Cτ̂a (f ) = T r(f ) = (i~)d R2d d! and the statement follows.  Given above identifications of Cτ̂a and Cτ̂t , the theorem 2.27 implies the following result.
Theorem 2.32 (Algebraic Index Theorem). Suppose a ∈ CC0per A~c is a cycle, then Z X   −d up . HKR(σ(a)) Â(TC M)eθ T r(a) = u M p≥0 2p 3. Equivariant Gelfand-Fuks map Suppose Γ is a discrete group acting by automorphisms on A~ . Then we can extend fr as follows. First note that the action on A~ induces an action on (M, ω). the action to M fr and γ ∈ Γ, then let γ (m, ϕm ) = (γ (m) , ϕγ ), here ϕγ is given Now suppose (m, ϕm ) ∈ M m m by ~ (M) \ ~ A\ γ(m) −→ A (M)m −→ W, where the first arrow is given by the action of Γ on A~ and the second arrow is given by ϕm . br acts on M fr by postcomposition and Γ acts on M fr by precomposition, Note that, since G we find that the two actions commute. It should be apparent from the preceding section that we will need an equivariant version of the Gelfand-Fuks map in the deformed setting. This will allow us to derive the EQUIVARIANT ALGEBRAIC INDEX THEOREM 15 equivariant algebraic index theorem from the Lie algebraic one. To do this we will extend the definition of the one-form AF to the Borel construction EΓ ×Γ M. Explicitly this is done by defining connection one-forms AF k on the manifolds ∆k × Γk × FM which satisfy certain boundary conditions. This connection one-form will then serve the usual purpose in the Gelfand-Fuks map, only now the range of the map will be a model for the equivariant cohomology of the manifold. Assume that Γ is a discrete group acting on a manifold X. Set Xk := X × Γk . Define the face maps ∂ik : Xk → Xk−1 by  −1  if i = 0 (γ1 (x), γ2 , . . . , γk ) k ∂i (x, γ1 , . . . , γk ) = (x, γ1 , . . . , γi γi+1 , . . . , γk ) if 0 < i < k  (x, γ , . . . , γ ) if i = k 1 k−1 We denote the standard k-simplex by ( ∆k := and define by ǫki : ∆k−1 → ∆k (t0 , . . . , tk ) ≥ 0| k X ti = 1 i=0 ) ⊂ Rk+1 ( (0, t0 , . . . , tk−1 ) if i = 0 ǫki (t0 , . . . , tk−1) = (t0 , . . . , ti−1 , 0, ti , . . . , tk−1 ) if 0 < i ≤ k Definition 3.1. A de Rham-Sullivan, or compatible form ϕ of degree p is a collection of forms ϕk ∈ Ωp (∆k × Xk ), k = 0, 1, . . ., satisfying (8) (ǫki × id)∗ ϕk = (id ×∂ik )∗ ϕk−1 ∈ Ωp (∆k−1 × Xk ) for 0 ≤ i ≤ k and any k > 0. If ϕ = {ϕk } is a compatible form, then dϕ := {dϕk } is also a compatible form; for two compatible forms ϕ = {ϕk } and ψ = {ψk } their product ϕψ := {ϕk ∧ ψk } is another compatible form. We denote the space of de Rham-Sullivan forms by Ω• (M ×Γ EΓ) in view of the following Theorem 3.2. We have H• (Ω• (M ×Γ EΓ), d) ≃ H•Γ (M) where the left hand side is the cohomology of the complex Ω• (M ×Γ EΓ) and the right hand side is the cohomology of the Borel construction M ×Γ EΓ. See for instance [8] for the proof. More generally, let V be a Γ-equivariant bundle on X. Let πk : Xk → X be the projection and let Vk := πk∗ V . Notice that we have canonical isomorphisms (9) (∂ik )∗ Vk−1 ∼ = Vk . 16 ALEXANDER GOROKHOVSKY, NIEK DE KLEIJN, RYSZARD NEST Definition 3.3. Let V be Γ-equivariant vector bundle. A V valued de Rham-Sullivan (compatible) form ϕ is a collection ϕk ∈ Ωp (∆k × Xk ; Vk ), k = 0, 1, . . ., satisfying the conditions (8), where we use the isomorphisms (9) to identify (∂ik )∗ Vk−1 with Vk . We let Ω• (M ×Γ EΓ; V ) denote the space of V -valued de Rham-Sullivan (compatible) forms. For equivariant vector bundles V and W there is a product Ω• (M ×Γ EΓ; V ) ⊗ Ω• (M ×Γ EΓ; W ) −→ Ω• (M ×Γ EΓ; V ⊗ W ) defined as for the scalar forms by ϕψ := {ϕk ∧ ψk }. Assume that we have a collection of connections ∇k on the bundles Vk satisfying the compatibility conditions (ǫki × id)∗ ∇k = (id ×∂ik )∗ ∇k−1 . (10) Then for a compatible form ϕ = {ϕk } ∇ϕ := {∇k ϕk } is again a compatible form. Notation 3.4. Now let M be a symplectic manifold and Γ a discrete group acting by symplectomorphisms on M. We introduce the following notations: PΓk := ∆k × (FM )k = ∆k × FM × Γk , and similarly MΓk := ∆k × Mk = ∆k × M × Γk and fk := ∆k × (Mr )k = ∆k × M fr × Γk . M Γ Note that PΓk → MΓk is a principal Sp(2d)-bundle, namely the pull-back of FM → M via fk is the pull-back of M fr → M. We define Ω• (M k ; L) the obvious projection. Similarly M Γ Γ for a (g, Sp(2d))-module L as we did for M above only replacing FM by PΓk and considering b1 -principal the trivial action of the symplectic group on ∆k × Γk . We shall denote the G fr → FM by π1 , the G br -principal bundle M fr → M by πr and the Sp(2d)-principal bundle M bundle FM → M by π. fr )k over ∆k−1 × (FM )k Note that, for all i and k, the (obvious) fibration of ∆k−1 × (M fk−1 over P k−1. is canonically isomorphic to the pull back by id ×∂ik of the fibration of M Γ Γ fk−1 → P k−1, there is a natural pull back Hence, for any section F of the projection M Γ Γ fr )k → ∆k−1 × (FM )k making the following (id ×∂ik )∗ F, the unique section of ∆k−1 × (M diagram commutative: (11) fr )k ∆k−1 × (M id ×∂ik O fk−1 . M Γ / O (id ×∂ik )∗ F ∆k−1 × id ×∂ik (FM )k / F PΓk−1 EQUIVARIANT ALGEBRAIC INDEX THEOREM 17 Lemma 3.5. There exist sections : Fk fk M Γ  PΓk fk → P k satisfying the compatibility conditions of the projections M Γ Γ (ǫki × id)∗ Fk = (id ×∂ik )∗ Fk−1 Proof. We construct the sections Fk recursively. In section 2 we constructed the inital section fr . Set F0 := Fr . Fr : PΓ0 = FM −→ M Now suppose we have found Fl satisfying the compatibility conditions for all l < k. fk → P k is trivial, and thus, by fixing a trivialization, Notice that the principal bundle M Γ Γ b1 . which we identify with we can view its sections as functions on PΓk with values in G a vector space g1 via the exponential map. The compatibility conditions require that Fk b1 can be takes on certain values determined by Fk−1 on (∂∆k ) × FM × Γk ⊂ PΓk . Since G identified with a vector space g1 via the exponential map, Fk can be extended smoothly from (∂∆k ) × F × Γk to PΓk  As before we can construct the sections in the lemma above Sp(2d) equivariantly and we will fix a system of such equivariant sections {Fk }k≥0 from now on. Now, as before, fk (which we can use the sections Fk to pull back the canonical connection form from M Γ f f was itself pulled back from M through the composition of the projection onto Mr and an inclusion), to define a g-valued differential form AF k on PΓk for each k.
b r )-cochain complex. Then we denote by L• , ∂L Notation 3.6. Suppose (L• , ∂L ) is a (g, G πr fr , i.e. with total space M fr × b L. the bundle of cochain complexes over M associated to M Gr We will denote the pull-back to the MΓk by the same symbol. Note that the pullback fr → FM with fiber the (g, G b1 )-cochain complex π ∗ Lπr is exactly the bundle associated to M b1 ֒→ G b r , i.e the pull-back has total space M fr × b L. Thus we will denote L given by G G1 ∗ π Lπr = Lπ1 . Again we will use the same notation for the pull-backs over the PΓk . br -bundle M fr → M we find that Γ also acts Remark 3.7. Note that since Γ acts on the G • fr → FM , we find on Lπr and, since this action lifts to an Sp(2d)-equivariant action on M • • a corresponding action on the space Ω (M; L ). Let us be a bit more precise about this fr → FM . This means also action. Note first that the section F0 yields a trivialization of M that it yields a trivialization (denoted by the same symbol) ∼ f F0 : FM × L −→ M r ×G1 L 18 ALEXANDER GOROKHOVSKY, NIEK DE KLEIJN, RYSZARD NEST explicitly given by (p, ℓ) 7→ [F0 (p), ℓ] with the inverse given by mapping [ϕm , ℓ] to (π1 (ϕm ), (ϕm ◦ F0 (π1 (ϕm ))−1 ) (ℓ)). In these terms the action is given by γ(η ⊗ ℓ) = (γ ∗ η) ⊗ (F0−1 )∗ γ ∗ F0∗ ℓ where (F0−1 )∗ γ ∗ F0∗ ℓ is the section given by p 7→ (γ(p), F0 (p)γF0 (γ(p))−1 ℓ).
We consider the corresponding action of Γ on the spaces Ω• MΓk ; L• , where we use Fk instead of F0 (or in fact on Ω• (N × M; L• ) for any N).
Note that the differential forms AFk define flat connections ∇
Fk on Ω• MΓk ; L• for Q all k and so we can consider the product complex k Ω• MΓk ; L• with the differential Q ˜ + ∂L , where ∇ ˜ = ∇ k ∇Fk . Note also that the connections ∇Fk satisfy the compatibility conditions of the equation (10). Now we can consider the equivariant Gelfand–Fuks map Y
• GFΓ : CLie (g, sp (2d) ; L• ) −→ Ω• MΓk ; L• k given by (12) GFΓ (χ)k = χ ◦ A⊗p Fk , p where χ ∈ CLie (g, sp (2d) ; L• ) and the subscript k refers to taking the k-th coordinate in the product. In other words the definition is the same as in definition 2.16 only we now use the compatible system of connections AFk . p Lemma 3.8. For all χ ∈ CLie (g, sp(2d); L• ) we have that GFΓ (χ) ∈ Ω• (M ×Γ EΓ, L• ). Proof. The boundary conditions put on the sections Fk in lemma 3.5 are meant exactly to ensure this property of the Gelfand-Fuks map GFΓ . The lemma follows straightforwardly from these boundary conditions.  Theorem 3.9. The equivariant Gelfand-Fuks map is a morphism of complexes • GFΓ : CLie (g, sp (2d) ; L• ) → Ω• (M ×Γ EΓ, L• ) . Proof. This proof is exactly the same as in the non-equivariant setting, carried out coordinate-wise in the product. Since we do not give the usual proof in this article let us be a bit more explicit. Note that Y
• GFΓ : CLie (g, sp (2d) ; L• ) −→ Ω• MΓk ; L• k is given by   GFΓ (χ)mk (X1 , . . . Xp ) = χ (AFk )mk X1 , . . . , (AFk )mk Xp , EQUIVARIANT ALGEBRAIC INDEX THEOREM 19 p for χ ∈ CLie (g, sp (2d) ; L• ), mk ∈ PΓk and Xi ∈ Tmk PΓk . The differential ∂L and GFΓ clearly commute so it is left to show that ˜ ◦ GFΓ . GFΓ ◦ ∂Lie = ∇ This follows by direct computation using the facts that 1 dAFk + [AFk , AFk ] = 0 2 and ˜ = ∇ Y dPΓk + AFk , k here dPΓk refers to the de Rham differential on PΓk .  4. Pairing with HCper A~c ⋊ Γ •
In order to derive the equivariant version of the algebraic index theorem we should show that the universal class τ̂a maps to the class of the trace supported at the identity under the equivariant Gelfand-Fuks map GFΓ constructed in the previous section. The class τ̂a br )-cochain complex lives in the Lie algebra cohomology with values in the (g, G L• := Hom−• (CC•per (W~ ), Ω̂−• [u−1 , uK[~−1 , ~K[2d]). Here the action is induced (through conjugation) by the action on W and by the action (by modding out ~) on Ω̂. From now on the notation L• will refer to this complex. The differential ∂L on L• is given by viewing it as the usual morphism space internal to chain complexes. In order to derive the equivariant algebraic index theorem we shall have to pair classes in Lie algebra cohomology with values in L• with periodic cyclic chains of A~c ⋊ Γ using the equivariant Gelfand–Fuks map. Since the trace on A~c ⋊ Γ is supported at the identity we only need to consider the component of the cyclic complexes supported at the identity. Definition 4.1 (Homogeneous Summand). Let CC•per (A~ ⋊ Γ)e be the subcomplex spanned (over C[u−1 , uK) by the chains a0 γ0 ⊗ . . . ⊗ an γn such that γ0 γ1 . . . γn = e ∈ Γ, where e denotes the neutral element of Γ. Proposition 4.2. The map D : CC•per (A~ ⋊ Γ) −→ C• (Γ; CC•per (A~ )) given by composing the projection CC•per (A~ ⋊ Γ) −→ CC•per (A~ ⋊ Γ)e with the quasi-isomorphism of theorem A.13 in the appendix is a morphism of complexes. 20 ALEXANDER GOROKHOVSKY, NIEK DE KLEIJN, RYSZARD NEST The proof is contained in the appendix. As in lemma 2.28, the canonical inclusion ι  
Ω• (M ×Γ EΓ)[~−1 , ~K[u−1 , uK[2d], d → Ω• (M ×Γ EΓ; Ω̂−• [~−1 , ~K[u−1, uK[2d]), ∇F + dˆ is a quasi-isomorphism and we will denote its quasi-inverse by T T : Ω• (M ×Γ EΓ; Ω̂−• [~−1 , ~K[u−1, uK[2d]) −→ Ω• (M ×Γ EΓ)[~−1 , ~K[u−1 , uK[2d] Definition 4.3. We define the pairing h·, ·i : Ω• (M ×Γ EΓ; L• ) × C• (Γ; CC•per (A~c (M))) −→ C[~−1 , ~K[u−1, uK per as follows. In the following let α = a ⊗ (g1 ⊗ g2 ⊗ . . . ⊗ gp ) ∈ CCk−p (A~ (M)) ⊗ (CΓ)⊗p and let ϕ ∈ Ω• (M ×Γ EΓ; L• ). We define Z hϕ, αi := T ϕp (JF∞p (a)), ∆p ×M ×g1 ×...×gp is the map given by taking the ∞-jets of elements of A~ (M) relative to the where Fedosov connection ∇Fp as in the example 2.18 using the section Fp over MΓk . Since the integral of ξ ∈ Ωk (M ×Γ EΓ) over any simplex ∆p for p > k will vanish, the pairing h·, ·i extends by linearity to C• (Γ; CC•per (A~c (M))). JF∞p Lemma 4.4. We have: ˜ F + ∂L )ϕ, αi = (−1)|ϕ|+1 hϕ, (δΓ + b + uB)αi h(∇ Proof. Z ˜ F + ∂L )ϕ, αi = h(∇ ˜ F + ∂L )ϕp )(J ∞ (a)) T ((∇ Fp ∆p ×M ×g1 ×...×gp ˆ p )(J ∞ (a)) − (−1)|ϕ| ϕp (J ∞ ((b + uB)a)). Also, since Notice that (∂L ϕp )(JF∞p (a)) = d((ϕ Fp Fp ∞ ∞ ∞ ˜ ˜ ˜ ∇F (JFp (a)) = 0 we have (∇F ϕp )(JFp (a)) = ∇F (ϕp )(JFp (a)). Combining these formulas we ˜ F + ∂L )ϕ, αi equals obtain that h(∇ Z   ˆ p (J ∞ (a)) − (−1)|ϕ| ϕp (J ∞ ((b + uB)a))) = ˜ F + d)(ϕ (13) T (∇ Fp Fp ∆p ×M ×g1 ×...×gp Z dT (ϕp (JF∞p (a)) − (−1)|ϕ| hϕ, (b + uB)αi ∆p ×M ×g1 ×...×gp Applying Stokes’ formula to R ∆p ×M ×g1 ×...×gp dT (ϕp (JF∞p (a)) and noticing that the collection of forms {T (ϕp (JF∞p (a))} is compatible we see that Z (14) dT (ϕp (JF∞p (a)) = (−1)|ϕ|+1hϕ, δΓ αi ∆p ×M ×g1 ×...×gp EQUIVARIANT ALGEBRAIC INDEX THEOREM 21 The statement of the lemma now follows from (13) and (14).  Recall that we have a cap-product ∩ C• (Γ; CC•per (A~c (M))) ⊗ C • (Γ, C) −→ C• (Γ; CC•per (A~c (M))). Definition 4.5. Let ξ ∈ C • (Γ, C) be a cocycle. Define • •+|ξ| Iξ : CLie (g, sp(2d); L• ) −→ CCper (A~c ⋊ Γ) by Iξ (λ)(a) = ǫ(|λ|)hGFΓ (λ), D(a) ∩ ξi for all λ ∈ CLie (g, sp(2d); L ) and a ∈ CC•per (A~c ⋊ Γ), where • • ǫ(m) = (−1)m(m+1)/2 . Proposition 4.6. The map • •+|ξ| Iξ : (CLie (g, sp(2d); L• ), ∂Lie + ∂L ) −→ CCper (A~c ⋊ Γ), (b + uB)∗ is a morphism of complexes.
Proof. Using Theorem 3.9 and Lemma 4.4 we have Iξ ((∂Lie + (−1)r ∂L )λ))(a) = ǫ(|λ| + 1)hGFΓ ((∂Lie + (−1)r ∂L )λ)), D(a) ∩ ξi = ˜ F +∂L )GFΓ (λ), D(a)∩ξi = (−1)|λ|+1 ǫ(|λ|+1)hGFΓ(λ), (δΓ +b+uB)(D(a))∩ξi = ǫ(|λ|+1)h(∇ ǫ(|λ|)hGFΓ (λ), (D((b + uB)a)) ∩ ξi = Iξ (λ)((b + uB)a) and the statement follows.  •+|ξ| Remark 4.7. The induced map on cohomology Iξ : H (g, sp(2d); L• ) −→ HCper (A~c ⋊ Γ) is easily seen to depend only on the cohomology class [ξ] ∈ H• (Γ, C). • 5. Evaluation of the equivariant classes In the previous sections we defined the map Iξ : H0 (g, sp(2d); L• ) −→ HCkper (A~c ⋊ Γ), where k = |ξ|. The last step in proving the main result of this paper is to evaluate the classes appearing in Lie algebraic index theorem 2.27. First of all we consider the image under Iξ of the trace density τ̂a . Consider the map hGFΓ (τ̂a ), ·i : C0 (Γ; C0 (A~c )) −→ C[~−1 , ~K. Since in degree 0 the equivariant Gelfand–Fuks map is given by the ordinary Gelfand–Fuks map on M, this map coincides with the canonical trace T r (cf. the proof of theorem 2.32). It follows that hGFΓ (τ̂a ), α ⊗ (γ1 ⊗ . . . γk ) ∩ ξi = ξ(γ1, . . . , γk )T r(α) From this discussion we obtain the following: 22 ALEXANDER GOROKHOVSKY, NIEK DE KLEIJN, RYSZARD NEST Proposition 5.1. We have Iξ (τ̂a ) = T rξ where T rξ is a cocycle on A~c (M) ⋊ Γ given by (15) T rξ (a0 γ0 ⊗ . . . ⊗ ak γk ) = ξ(γ1 , . . . , γk )T r(a0 γ0 (a1 ) . . . (γ0 γ1 . . . γk−1 (ak )) if γ0 γ1 . . . γk = e and 0 otherwise. Definition 5.2. The equivariant Weyl curvature θΓ is defined as the image of θ̂ under GFΓ followed by (CJ~K-linear extension of) the map in Theorem 3.2. Similarly, the equivariant Â-genus of M, denoted Â(M)Γ , is defined as the image of  under the equivariant Gelfand– Fuks map followed by (CJ~K-linear extension of) the isomorphism in Theorem 3.2. Example 5.3. Let us provide an example of the characteristic class θΓ . To do this consider the example of group actions on deformation quantization given in [14]. Namely, we consider the symplectic manifold R2 /Z2 = T2 , the 2-torus, with the symplectic structure ω = dy ∧ dx induced from the standard one on R2 , where x, y ∈ R/Z are the standard coordinates on T2 . We then consider the action of Z on T2 by symplectomorphisms where the generator of Z acts by T : (x, y) 7→ (x+x0 , y +y0). Note that, for a generic pair (x0 , y0 ), the quotient space is not Hausdorff. The Fedosov connection ∇F given as in Example 2.12 descends to the connection on T2 which is, moreover, Z-invariant (where we endow C ∞ (T2 , W) with the action of Z induced by the symplectic action on T2 ). It follows that A~ = Ker ∇F is a Z-equivariant ω deformation with the characteristic class i~ . We can obtain a more interesting example by modifying the previous one as follows (cf. [14]). Let u ∈ C ∞ (T2 , W) be an invertible element such that u−1 (∇F u) is central. Define a new action of Z on C ∞ (T2 , W) where the generator acts by w 7→ u−1 (T w)u. Ker ∇F is again invariant under this action and we thus obtain an action of Z on A~ . To describe its characteristic class note that, since EZ ∼ = R, we find that the cohomology • • • 2 2 ∼ 3 H (T ) = H (R × T ) H (T ). Let ν be a compactly supported 1-form on R with = Z RZ ν = 1. Denote by τ the translation t → t − 1. Then R X α̃ = (τ ∗ )n (ν) ∧ (T ∗ )n (U −1 ∇F U) n∈Z is a Z-invariant form on R × T2 , hence a lift of a form, say α, on R ×Z T2 = T3 . The characteristic class of the associated Z-equivariant deformation is equal to ω θZ = + α. i~ Finally we arrive at the main theorem of this paper. Let R : HΓeven (M) → HΓ• (M)[u] be given by R(a) = udeg a/2 a and recall the morphism defined in (19) • Φ : HΓ• (M) −→ HCper (Cc∞ (M) ⋊ Γ). 23 Theorem 5.4 (Equivariant Algebraic Index Theorem). Suppose a ∈ CC0per (A~c ⋊ Γ) is a cycle, then we have D     E θΓ T rξ (a) = Φ R Â(M)Γ e [ξ] , σ(a) • where h·, ·i denotes the pairing of CCper and CC•per . Proof. The theorem follows from Theorem 2.27 by applying the morphism Iξ . The image of τa under Iξ is T rξ (cf. Proposition 5.1). On the other hand, by equation (19), " !#      X Iξ Âf eθ̂ up τ̂t = Φ R Â(M)Γ eθΓ [ξ] . 2p p≥0  Note that the form of the theorem 1.4 stated in the introduction follows by considering the pairing of periodic cyclic cohomology and K-theory using the Chern–Connes character [15]. Appendices Below we shall fix our conventions with regard to cyclic/simplicial structures and homologies. We will also define the complexes we use to describe group (co)homology, Lie algebra cohomology and cyclic (co)homology. The general reference for this section is [15]. Fix a field k of characteristic 0. Appendix A. Cyclic/simplicial structure Let Λ denote the cyclic category. Instead of giving the intuitive definition let us simply give a particularly useful presentation. The cyclic category Λ has objects [n] for each n ∈ Z≥0 and is generated by δin ∈ Hom([n − 1], [n]) and σin ∈ Hom([n + 1], [n]) for 0 ≤ i ≤ n for all n ∈ Z≥0 tn ∈ Hom([n], [n]) with the relations n−1 δjn ◦ δin−1 = δin ◦ δj−1 if i < j n+1 σjn ◦ σin+1 = σin ◦ σj+1 if i ≤ j n−1 σjn ◦ δin+1 = δin ◦ σj−1 if i < j σjn ◦ δin+1 = Id[n] if i = j, j + 1 σjn ◦ δin+1 tn ◦ δin tn ◦ σin = = = n ◦ σjn−1 δi−1 n δi+1 ◦ tn−1 n σi+1 ◦ tn+1 if i > j + 1 if 0 ≤ i < n if 0 ≤ i < n tn+1 n tn ◦ δnn tn ◦ σnn = Id[n] = δ0n = σ0n ◦ tn+1 ◦ tn+1 . 24 Using only the generators δin and σin and relations not involving tn ’s gives a presentation of the simplicial category △. A contravariant functor from Λ (△) to the category of kmodules is called a cyclic (simplicial) k-module. Definition A.1. Given a unital associative k-algebra A we shall denote by A♮ the functor Λop → k − Mod given by A♮ ([n]) = A⊗n+1 and δin (a0 ⊗ . . . ⊗ an ) =a0 ⊗ . . . ⊗ ai ai+1 ⊗ . . . ⊗ an if 0 ≤ i < n δnn (a0 ⊗ . . . ⊗ an ) =an a0 ⊗ a1 ⊗ . . . ⊗ an−1 σin (a0 ⊗ . . . ⊗ an ) =a0 ⊗ . . . ⊗ ai ⊗ 1 ⊗ ai+1 ⊗ . . . ⊗ an for all 0 ≤ i ≤ n tn (a0 ⊗ . . . ⊗ an ) =a1 ⊗ . . . ⊗ an ⊗ a0 Note that if A admits a group action of the group G by unital algebra homomorphisms then G also acts on A♮ (diagonally). Definition A.2. Given a group G we shall denote by Gk♮ the functor Λop → k − Mod given by Gk♮ ([n]) = (kG)⊗n+1 and δin (g0 ⊗ . . . ⊗ gn ) =g0 ⊗ . . . ⊗ gˆi ⊗ . . . ⊗ gn for all 0 ≤ i ≤ n σin (g0 ⊗ . . . ⊗ gn ) =g0 ⊗ . . . ⊗ gi ⊗ gi ⊗ gi+1 ⊗ . . . ⊗ gn for all 0 ≤ i ≤ n tn (g0 ⊗ . . . ⊗ gn ) =g1 ⊗ g2 ⊗ . . . ⊗ gn ⊗ g0 . Note that G acts on Gk♮ from the right by g · (g0 ⊗ . . . ⊗ gn ) = g −1 g0 ⊗ . . . ⊗ g −1 gn . Definition A.3. Given two cyclic k-modules A♮ and B ♮ we shall denote by A♮B the cyclic k-module given by A♮B([n]) = A♮ ([n]) ⊗ B ♮ ([n]) with the diagonal cyclic structure. A.1. Cyclic homologies. Given a cyclic k-module M ♮ we can consider four different complexes associated to the simplicial/cyclic structure. To define them we shall first define two operators: b and B. The first is induced through the Dold–Kan correspondence and uses only the simplicial structure. It is given by bn = n X (−1)i δin : M ♮ ([n]) −→ M ♮ ([n − 1]). i=0 By using the simplicial identities above it is easily verified that bn−1 bn = 0. To define the “Hochschild” complex it is enough to have just the operators bn . To define the three cyclic complexes we shall use the operator ! n X n n Bn = (t−1 (−1)in tin : M ♮ ([n]) −→ M ♮ ([n + 1]). n+1 + (−1) ) ◦ σn ◦ i=0 25 Note that Bn+1 Bn = 0 since (16) n+1 X (−1)i(n+1) tin+1 ◦ (t−1 n+1 n + (−1) ) = i=0 n+1 X i−1 (−1)i(n+1) (tn+1 + (−1)n tin+1 ) = 0. i=0 Vanishing of the above expression follows since the sum telescopes except for the first term n(n+1) n+1 t−1 tn+1 , which also cancel each other. Note also that n+1 and the last term −(−1) (17) bn+1 Bn + Bn−1 bn = 0, this can be seen by writing out both operators as sums of operators in the normal form δkn σln−1 tin . From now on we will drop the subscripts of the b and B operators. The cyclic module M ♮ gives rise to a graded module {Mn♮ }n∈Z≥0 by Mn♮ = M ♮ ([n]). Then we see that the operator b turns M ♮ into a chain complex. Definition A.4. The Hochschild complex (C•Hoch (M ♮ ), b) of the cyclic module M ♮ is defined as CnHoch (M ♮ ) := M ♮ ([n]) equipped with the boundary operator b (of degree −1). The corresponding homology shall be denoted HH• (M ♮ ). Note that we have not used the full cyclic structure of M ♮ to construct
the Hochschild △ complex. In fact one can form the Hochschild complex C• (M ), b of any simplicial k-module M △ in exactly the same way. Note that by (17) and (16) we find that (b + B)2 = 0. This implies that we could consider a certain double complex with columns given by the Hochschild complex. Note however that, if b is of degree −1 on the Hochschild complex, the operator B is naturally of degree +1. We can consider a new grading for which the operator b + B is of homogeneous degree −1. In order to make this grading easy to see, it will be useful to introduce the formal variable u of degree −2. This leads us to several choices of double complexes. Definition A.5. We define the cyclic complex by  .  Hoch ♮ −1 (CC• (M ♮ ), δ ♮ ) := C• (M )[u , uK C Hoch (M ♮ )JuK , b + uB , • the negative cyclic complex by ♮ (CC•− (M ♮ ), δ− ) := C•Hoch (M ♮ )JuK, b + uB and finally the periodic cyclic complex by

♮ (CC•per (M ♮ ), δper ) := C•Hoch (M ♮ )[u−1 , uK, b + uB . Here u denotes a formal variable of degree −2. The corresponding homologies will be per ♮ ♮ denoted HC• (M ♮ ), HC− • (M ) and HC• (M ) respectively. The cyclic cochain complexes • CCper (M ♮ ), CC−• (M ♮ ) and CC • (M ♮ ) are defined as the k-duals of the chain complexes. We shall often omit the superscripts ♮ when there can be no confusion as to what the cyclic structures are. 26 Remark A.6. Note that every “flavor” of cyclic homology comes equipped with spectral sequences induced from the fact that they are realized as totalizations of a double complex. The double complex corresponding to cyclic homology is bounded (second octant) and therefore the spectral sequence which starts by taking homology on columns converges to HC• . The negative (or periodic) cyclic double complex is unbounded, but concentrated in the (second,) third, fourth and fifth octant. This means that the spectral sequence per starting with taking homology in the columns converges again to HC− • (or HC• ). Note however that in this case the negative (or periodic) cyclic homology is given by the product totalization. The remark A.6 provides the proof of the following proposition. Proposition A.7. Suppose M ♮ and N ♮ are two cyclic k-modules and ϕ : N ♮ −→ M ♮ is a map of cyclic modules that induces an isomorphism on Hochschild homologies. Then ϕ induces an isomorphism on cyclic, negative cyclic and periodic cyclic homologies as well. Proof. The proof follows since ϕ induces isomorphisms on the first pages of the relevant spectral sequences, which converge.  A.2. Replacements for cyclic complexes. It will often be useful to consider different complexes that compute the various cyclic homologies. We shall give definitions of the complexes that are used in the main body of the article here. A.2.1. Crossed product. Suppose A is a unital k-algebra and G is a group acting on the left by unital algebra homomorphisms. We denote by A ⋊ G the crossed product algebra given by A ⊗ kG as a k-vector space and by the multiplication rule (ag)(bh) = ag(b)gh for all a, b ∈ A and g, h ∈ G. Note that the cyclic structure of (A ⋊ G)♮ splits over the conjugacy classes of G. Namely, given a tensor a0 g0 ⊗ a1 g1 ⊗ . . . ⊗ an gn , the conjugacy class of the product g0 · . . . · gn is invariant under δin , σin and tn for all i and n. So we have M (A ⋊ G)♮ = (A ⋊ G)♮x x∈hGi where we denote the set of conjugacy classes of G by hGi and the span of all tensors a0 g0 ⊗ . . . ⊗ an gn such that g0 · . . . · gn ∈ x by (A ⋊ G)♮x . The summand (A ⋊ G)♮e , here e = {e} the conjugacy class of the neutral element, is called the homogeneous summand. We shall use the specialized notation A♮G := A♮ ♮Gk♮ . Note that A♮G carries a right G action given by the diagonal action (the left action on A is converted to a right action by . inversion, i.e. G ≃ Gop ). Thus the co-invariants (A♮G)G = A♮G ha − g(a)i form another cyclic k-module. Proposition A.8. The homogeneous summand of (A ⋊ G)♮ is isomorphic to the coinvariants of A♮G. ∼ (A ⋊ G)♮e −→ (A♮G)G . 27 Proof. Consider the map given by a0 g0 ⊗. . .⊗an gn 7→ (g0−1 (a0 )⊗a1 ⊗g1 (a2 )⊗. . .⊗g1 . . . gn−1 (an ))♮(e⊗g1 ⊗g1 g2 ⊗. . .⊗g1 ·. . .·gn ), it is easily checked to commute with the cyclic structure and allows the inverse given by −1 −1 (a0 ⊗ . . . ⊗ an )♮(g0 ⊗ . . . ⊗ gn ) 7→ gn−1(a0 )gn−1g0 ⊗ g0−1 (a1 )g0−1g1 ⊗ . . . ⊗ gn−1 (an )gn−1 gn −1 this last tensor can also be expressed as gn−1a0 g0 ⊗ g0−1a1 g1 ⊗ . . . ⊗ gn−1 an gn .  Definition A.9. Suppose (M• , ∂) is a right kG-chain complex. Then we define the group homology of G with values in M as Q (C• (G; M), δ(G,M ) ) := Tot M• ⊗kG C•Hoch (G) where we consider the tensor product of kG-chain complexes with the obvious structure of left kG-chain complex on C•Hoch (G). Note that this means that Y Cn (G; M) = Mp ⊗kG CqHoch (G) p+q=n and δ(G,M ) = ∂ ⊗ Id + Id ⊗ b where we use the Koszul sign convention. Proposition A.10. Suppose M is a right kG-module. Then M ⊗ kG with the diagonal right action is a free kG-module. Proof. Let us denote the k-module underlying M by F (M), then F (M) ⊗ kG denotes the free (right) kG-module induced by the k-module underlying M. Consider the map M ⊗ kG −→ F (M) ⊗ kG given by m⊗g 7→ mg −1 ⊗g. It is obviously a map of kG-modules and allows for the inverse m ⊗ g 7→ mg ⊗ g.  Proposition A.11. Suppose F is a free right kG-module (we view it as a chain complex concentrated in degree 0 with trivial differential) then there exists a contracting homotopy HF : C• (G; F ) −→ C•+1 (G; F ). Suppose (F• , ∂) is a quasi-free right kG-chain complex (i.e. Fn is a free kG-module for all n) then the homotopies HFn give rise to a quasi-isomorphism ∼ ((F• )G , ∂) −→ (C• (G; F ), δ(G,F ) ). Proof. Note that F ≃ M ⊗ kG since it is a free module. So we find that Cp (G; F ) = (M ⊗ kG) ⊗kG (kG)⊗p+1 ≃ M ⊗ (kG)⊗p+1 by the map m ⊗ g ⊗ g0 ⊗ . . . ⊗ gp 7→ m ⊗ gg0 ⊗ . . . ⊗ ggp. Using this normalization we consider the map HM given by m ⊗ g0 ⊗ . . . ⊗ gp 7→ m ⊗ e ⊗ g0 ⊗ . . . ⊗ gp 28 and note that indeed p+1 p δG HM + HM δG = Id (we denote δG := δ(G,M ) = Id ⊗ b) for all p > 0. Now for the second statement we find that Fn ≃ Mn ⊗kG for each n since it is quasi-free. For each n we have the homotopy Hn := HFn given by the formula above on C• (G; Fn ). Then we consider the map QH : (Fp )G −→ Cp (G; F ) given by ∞ X q+1 1 (−H∂)q Hf QF ([f ]) = f − δG Hf + (−H∂)q f − ∂(−H∂)q−1 Hf − δG q=1 where we have dropped the subscript from H and we denote the class of f in the coinvariants FG by [f ] . One may check by straightforward computation that QF is a welldefined morphism of complexes. Now we note that the double complex defining C• (G; F ) is concentrated in the upper half plane and therefore comes with a spectral sequence with first page given by Hp (G; Fq ) which converges to H(Cp+q (G; F )) (group homology). Note however that since F• is quasi-free we find that Hp (G, Fq ) = 0 unless p = 0 and H0 (G, Fq ) = (Fq )G . Thus, since QF induces an isomorphism on the first page and the spectral sequence converges, we find that QF is a quasi-isomorphism.  As a kG-module we see that A♮G([n]) = A♮ ([n]) ⊗ Gk♮ ([n]) = B([n]) ⊗ kG with the diagonal action, where B([n]) = A⊗n+1 ⊗ kG⊗n . So by proposition A.10 we find that the Hochschild and various cyclic chain complexes corresponding to A♮G are quasi-free. Thus we can construct the quasi-isomorphisms from proposition A.11 for each chain complex associated to the cyclic module A♮G. So we find four quasi-isomorphisms which we shall denote QHoch , Q, Q− and Qper corresponding to the Hochschild, cyclic, negative cyclic and periodic cyclic complexes respectively. Proposition A.12. The map A♮G −→ A♮ given by (a0 ⊗ . . . ⊗ an )♮(g0 ⊗ . . . ⊗ gn ) 7→ a0 ⊗ . . . ⊗ an induces a quasi-isomorphism on all associated complexes. Proof. Note that, by proposition A.7, it is sufficient to prove the statement for the Hochschild complexes. Let us denote the standard free resolution of G by F (G), note that F (G) = (C•Hoch (Gk♮ ), b). The map given above is obtained by first applying the Alexander–Whitney map M CnHoch (A♮ ) ⊗ CnHoch (Gk♮ ) −→ CpHoch (A♮ ) ⊗ CqHoch (Gk♮ ), p+q=n 29 which yields a quasi-isomorphism ∼ C•Hoch (A♮G) −→ C•Hoch (A♮ ) ⊗ C•Hoch (Gk♮ ), where we consider the tensor product of chain complexes on the right-hand side. Then one simply takes the cap product with the generator in H ∗ (F (G)∗ ) ≃ k, which is also a quasi-isomorphism. So we find that the map is a quasi-isomorphism for the Hochschild complexes.  Note that the map given in proposition A.12 is also G-equivariant and therefore it induces a map C• (G; A♮G) −→ C• (G; A♮ ) which is a quasi-isomorphism when we consider the group homology complex with values in the various complexes associated to A♮ . Theorem A.13. The composite maps from the Hochschild and various cyclic complexes associated to (A⋊Γ)♮e to the group homology with values in the various Hochschild and cyclic complexes associated to A♮ implied by propositions A.8 and A.12 are quasi-isomorphisms, i.e. there are quasi-isomorphisms

∼ C•Hoch (A ⋊ G)♮e , b −→ C• (G; C•Hoch (A))

∼ CC• (A ⋊ G)♮e , δ ♮ −→ C• (G; CC•(A)) 
♮ ∼ CC•− (A ⋊ G)♮e , δ− −→ C• (G; CC•− (A)) and
♮
∼ CC•per (A ⋊ G)♮e , δper −→ C• (G; CC•per (A)). Remark A.14. Note that since the cyclic and Hochschild complexes are bounded below the product totalizations in our definition of group homology agrees with the (usual) direct sum totalizations. In the periodic cyclic and negative cyclic cases they do not agree in general. Remark A.15. Suppose that a discrete group Γ acts on a smooth manifold M by diffeomorphisms. The above produces a morphism of complexes CC•per (C ∞ (M)c ⋊ Γ) → C• (Γ, CC•per (Cc∞ (M)) Composing it with the morphism CC•per (Cc∞ (M)) −→ Ω•c (M)[u−1 , uK, induced by the map f0 ⊗ f1 ⊗ . . . ⊗ fn 7→ 1 f0 df1 . . . dfn n! we get a morphism of complexes (18) CC•per (C ∞ (M)c ⋊ Γ) → C• (Γ, Ω•c (M)[u−1 , uK). 30 In the case when M is oriented and the elements of Γ preserve orientation, the transpose of this map can be interpreted as a morphism of complexes • Φ : C • (Γ, Ωdim(M )−• (M)[u−1 , uK) −→ CCper (Cc∞ (M) ⋊ Γ), (19) compare [5] section 3.2.δ. A.2.2. Group Homology. It is often useful to consider instead of the above complex for group homology an isomorphic complex, which we will call the non-homogeneous complex. Definition A.16. Suppose (M• , ∂) is a right kG-chain complex, then we set Y C̃n (G; M) := Mq ⊗ (kG)⊗p . p+q=n We define the operators δip : M• ⊗ (kG)⊗p → M• ⊗ (kG)⊗p−1 by δ0p (m ⊗ g1 ⊗ . . . ⊗ gp ) := g1 (m) ⊗ g2 ⊗ . . . ⊗ gp δip (m ⊗ g1 ⊗ . . . ⊗ gp ) := m ⊗ g1 ⊗ . . . ⊗ gi gi+1 ⊗ . . . ⊗ gp for all 0 < i < p and finally δpp (m ⊗ g1 ⊗ . . . ⊗ gp ) := m ⊗ g1 ⊗ . . . ⊗ gp−1. We define (C̃• (G; M), δ̃(G,M ) ) to be the chain complex given by δ̃(G,M ) = ∂ ⊗ Id + Id ⊗ δG p where δG = Pp p i=0 δi . Proposition A.17. There is an isomorphism of chain complexes C• (G; M) −→ C̃• (G; M). Proof. Consider the map Cn (G; M) −→ C̃n (G; M), given by −1 m ⊗ g0 ⊗ . . . ⊗ gp 7→ g0 (m) ⊗ g0−1g1 ⊗ g1−1 g2 ⊗ . . . ⊗ gp−1 gp . Note that it commutes with the differentials and allows for the inverse given by m ⊗ g1 ⊗ . . . ⊗ gp 7→ m ⊗ e ⊗ g1 ⊗ g1 g2 ⊗ . . . ⊗ g1 · . . . · gp .  We will usually use this chain complex when dealing with group homology and thus we will drop the tilde in the main body of this article. 31 A.3. Lie algebra cohomology. In this section let us describe the Lie algebra cohomology. Although there are various deep relations between Lie algebra cohomology and cyclic and Hochschild homologies we have chosen to present the complex in a separate manner. One could compute the Lie algebra cohomology using a Hochschild complex, however in the relative case (which we need in this article) there are several subtleties that we would rather avoid by considering a different complex. Definition A.18. Suppose g is a Lie algebra over k, h ֒→ g a subalgebra and (M• , ∂) is a g-chain complex. Then we denote
p p CLie (g, h; Mq ) := Homh ∧ g/h, Mq . We define operators p p p+1 ∂Lie : CLie (g, h; Mq ) −→ CLie (g, h; Mq ) by p ∂Lie ϕ(X0 , . . . , Xp ) = + X p X (−1)i Xi ϕ(X0 , . . . , X̂i , . . . , Xp ) i=0 (−1)i+j ϕ([Xi , Xj ], X0 , . . . , X̂i , . . . , X̂j , . . . , Xp ) 0≤i<j≤p p+1 p where the hats signify omission. Note that ∂Lie ∂Lie = 0 and ∂Lie commutes with ∂ by assumption. Thus we can consider the totalization of the corresponding double complex. We will denote the corresponding hypercohomology by H•Lie (g, h; M). Remark A.19. We actually only consider the Lie algebra cohomology of infinite dimensional Lie algebras here. For these the complex above is not very useful. The Lie algebras we consider come with a topology (induced by filtration) however and so do the coefficients. Using this fact we can consider in the above not simply anti-symmetric linear maps, but continuous anti-symmetric linear maps from the completed tensor products. References [1] Michael Atiyah and Isadore Singer, The Index Of Elliptic Operators on Compact Manifolds, Bull. Amer. Math. Soc., 69, 422-433, 1963. [2] Paul Bressler, Ryszard Nest and Boris Tsygan, Riemann-Roch Theorems Via Deformation Quantization I Adv. Math. 167, 1–25, 2002. [3] Paul Bressler, Ryszard Nest and Boris Tsygan, Riemann-Roch Theorems Via Deformation Quantization II Adv. Math. 167, 26–73, 2002. [4] Alain Connes, Non-Commutative Differential Geometry, IHES Publ. Math., 62, 257–360, 1985. [5] Alain Connes, Non-commutative Geometry, Academic Press, San Diego, 1990. [6] Alain Connes and Henri Moscovici, Hopf Algebras, Cyclic Cohomology and the Transverse Index Theorem Commun. Math. Phys., 198, 199-246, 1998. [7] Paul Bressler, Alexander Gorokhovsky, Ryszard Nest and Boris Tsygan, Algebraic Index Theorem for Symplectic Deformations of Gerbes, Contemporary Mathematics vol. 546 (Non-commutative Geometry and Global analysis), 2011. [8] Johan Dupont, Curvature and Characteristic Classes, Springer, LNM 640, Heidelberg, 1978. 32 [9] Boris Fedosov, The Index Theorem for Deformation Quantization, in Boundary Value Problems, Schrödinger Operators, Deformation Quantization; Akademie, Advances in Partial Differential Equations, 319-333, Berlin, 1995. [10] Boris Fedosov, Deformation Quantization and Index Theory, Akademie Verlag, Berlin, 1st Edition, 1996, chapter 5 and 6. [11] Israel Gelfand, Cohomology of Infinite-Dimensional Lie Algebras. Some Questions in Integral Geometry, Lecture given at ICM, Nice, 1970. [12] Israel Gelfand and David Kazhdan, Certain Questions of Differential Geometry and the Computation of the Cohomologies of the Lie Algebras of Vector Fields, Soviet Math. Doklady, 12, 1367-1370, 1971. [13] Simone Gutt, Deformation Quantization: an Introduction, 3rd edition, HAL, Monastir, 2005. [14] Niek de Kleijn Extension and Classification of Group Actions on Formal Deformation Quantizations of Symplectic Manifolds, preprint, arXiv:1601.05048, 2016. [15] Jean-Louis Loday, Cyclic Homology, 2nd edition, Springer, GMW 301, Heidelberg, 1998. [16] Ryszard Nest and Boris Tsygan, Algebraic Index Theorem Commun. Math. Phys. 172, 223–262, 1995. [17] Ryszard Nest and Boris Tsygan, Formal Versus Analytic Index Theorems, Internat. Math. Res. Notices, 11, 1996. [18] Ryszard Nest and Boris Tsygan, Deformations of Symplectic Lie Algebroids, Deformations of Holomorphic Symplectic Structures, and Index Theorems, Asian J. Math. 5 (2001), no. 4, 599–635. [19] Denis Perrot, Rudy Rodsphon, An equivariant index theorem for hypo-elliptic operators arXiv:1412.5042 [20] Marcus Pflaum, Hessel Posthuma and Xiang Tang, An Algebraic Index Theorem for Orbifolds, Adv. Math., 210, 83-121, 2007. [21] Markus Pflaum, Hessel Posthuma and Xiang Tang, On the Algebraic Index for Riemannian Étale Groupoids, Lett. Math. Phys., 90, 287-310, 2009. [22] Anton Savin, Elmar Schrohe, Boris Sternin, Uniformization and index of elliptic operators associated with diffeomorphisms of a manifold. Russ. J. Math. Phys. 22 (2015), no. 3, 410–420. [23] Anton Savin, Elmar Schrohe, Boris Sternin, On the index formula for an isometric diffeomorphism. (Russian) Sovrem. Mat. Fundam. Napravl. *46 * (2012), 141–152; translation in J. Math. Sci. (N.Y.) 201:818829 (2014) [24] Anton Savin, Elmar Schrohe, Boris Sternin, The index problem for elliptic operators associated with a diffeomorphism of a manifold and uniformization. (Russian) Dokl. Akad. Nauk 441 (2011), no. 5, 593–596; translation in Dokl. Math. 84 (2011), no. 3, 846–849 [25] Anton Savin, Boris Sternin Elliptic theory for operators associated with diffeomorphisms of smooth manifolds , arXiv:1207.3017 [26] Norman Steenrod, The Topology of Fiber Bundles, Princeton University Press, Princeton, 1951.