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Duflo Isomorphism and Chern-Weil Theory

We explain how the distributional index of a transversally elliptic operator on a principal G-manifold P that is obtained by lifting a Dirac operator on P/G can serve as a link between the Duflo isomorphism and Chern-Weil forms.


Introduction
The theme of this article arises in the typical context of a closed evendimensional spin (or spin-c) manifold M . There one naturally has a principal G-bundle P → M , where G is some compact Lie group, and a Dirac operator D acting on sections of an equivariant vector bundle P × G E. By picking a connection on P , one can horizontally lift the Dirac operator on M to P . The resulting differential operatorD, which may operate on C ∞ (P ) ⊗ E, is non-elliptic. However, it is transversally elliptic in the sense of Atiyah and Singer [1, Def. 1.3, p. 7], which means in our situation thatD is elliptic in horizontal directions relative to the selected connection on P . Continuing with the idea of Atiyah and Singer, one has the distributional index [D] of D, which is a formal Z-linear combination of characters of G. It is a theorem of Atiyah and Singer that [D] is a genuine distribution on G [1, Thm. 2.2, p. 10].
More is true in our case. Owing to the free G-action on P , the distribution [D] is supported at the identity, and it can be identified as an element of the center Z(g) of the universal enveloping algebra generated by the Lie algebra g of G. Thus [D] is subject to (the inverse of) the Duflo isomorphism Duf : S(g) G → Z(g).
Here S(g) G denotes the G-invariant subalgebra of the symmetric algebra generated by g; it can be identified as the algebra of G-invariant distributions on g that are supported at the origin. So it makes sense to pair Duf −1 [D] with a G-invariant analytic function ϕ defined on some neighborhood of the origin in g. One can now ask whether the following equation holds: HereD is the index class of D;φ is the characteristic class on M obtained from ϕ by the Chern-Weil homomorphism; andM is the fundamental homology class of M determined by the spin structure. Our aim is to show that Equation (1) holds under a reasonable condition on the connection associated withD and if we restrict the domain ofD to the G-invariant subspace (C ∞ (P ) ⊗ E) G . Denote the restriction ofD by D M . Then Equation (1) explicitly shows how the distributional index of D M serves has a link between the Duflo isomorphism and the Chern-Weil forms.
In a sense this is just another manifestation of the Atiyah-Singer index theorem. Indeed, when ϕ is the constant polynomial 1, the pairing D ,M calculates the usual (graded) index of D.

Setup
Throughout this article, M denotes a closed oriented Riemannian manifold of dimension n. Let D be a geometric Dirac operator acting on the sections of an equivariant vector bundle where P is the total space of a principal bundle κ : P → M whose fibers are isomorphic to a compact Lie group G, and E is a finitedimensional G-vector space over C associated with a representation ν : G → Aut(E).
We denote by ν * : g → End(E) the Lie algebra representation induced by the differential of ν at the identity.
We assume that E is a graded Cl(n)-module. Regarding the irreducible ones, there is only one (up to isomorphism) if n is odd, and there are two if n is even. If we fix one over the other for the even case, then we may speak of the irreducible graded Cl(n)-module S. Under such circumstances, E is of the form where Cl(n) acts canonically on S and trivially on W .
The existence of a Dirac operator on the sections of P × G E implies that the vector bundle P × G E admits a Cl(M )-module structure. This means that E is a module over the Clifford algebra Cl(n) generated by the Euclidean space R n and that we have a bundle map such that, when restricted to each fiber, we have an isomorphism of algebras Let Γ(P × G E) denote the space of sections of P × G E. Our assumption that D is a geometric Dirac operator means that there is a Clifford connection ∇ on Γ(P × G E), relative to which D is locally of the form for the tangent bundle T M of M . Now let∇ be the covariant derivative for the trivial bundle P × E → P that agrees with ∇ under the usual identification of Γ(P × G E) with Γ(P × E) G = (C ∞ (P ) ⊗ E) G . The covariant derivative∇ determines a connection θ on P , which in turn determines the horizontal subspace at each tangent space of P . We define the liftD of D as the differential operator on Γ(P × E) = C ∞ (P ) ⊗ E that is locally of the form The connection θ induces a G-invariant metric on P in the following way: Let , M be the pullback of the metric of M along the bundle projection κ, and let , g be an inner product on g that is invariant under the adjoint action of G on g. Then we define the metric , on P by With the metric on P at hand, we have the Riemannian connection ∇ P for X(P ) := Γ(T P ). For each X ∈ g, denote the fundamental vector field it generates on P byX. Letξ be a horizontal vector field. The vector field ∇ P Xξ is again horizontal; denote itξ ′ . Let ξ := κ * ξ , where κ * is the pushforward indued by κ, and denote its value at This gives a Lie algebra representation In terms of α x , the condition for the covariant derivative∇ on C ∞ (P ) ⊗ E to be a Clifford connection is that holds for all X ∈ g, ξ ∈ T x M , and x ∈ M . It is appropriate to call this as the Clifford condition for∇. If this condition holds, then the family {α x } x∈M of Lie algebra representations, together with the bundle map (3), defines collectively a Lie algebra representation This representation is equivalent to a Lie algebra representation where the two representations are related by Here the bracket on the right-hand side denotes commutation in Cl(n).
In terms of the Lie algebra representation γ, the Clifford condition (4) can be written as . This commutation relation implies that the g-action on E respects the factorization (2), so that ν * = γ ⊗ 1 + 1 ⊗ τ for some Lie algebra representation Remark. The Clifford condition (4) is always satisfied if the Lie algebra representation γ is induced by a Lie group representation G → Spin(n); in that case we have T M ∼ = P × G R n , where G acts on R n through the double covering Spin(n) → SO(n).

Distributional Index
The lifted operatorD on P is by construction a transversally elliptic operator on E-valued functions on P . Because the connection θ is G-invariant, D is also G-invariant, so the kernel ofD is a G-space. Denote the even and odd parts of the kernel as ker(D + ) and ker(D − ), respectively. LetĜ be the unitary dual of G. It is a result of Atiyah and Singer [1, Lem. 2.3, p. 10] that the pairings are well-defined elements of the formal representation group Moreover, identifying [V ] with its character, the formal sums [kerD ± ] converge in the distributional sense [1, Thm. 2.2, p. 10]. The distributional index ofD is then defined as Our real interest, however, lies in the "trivial part" of [D], that is, the distributional index [D M ] where D M is the restriction ofD onto the G-invariant subspace (C ∞ (P )⊗E) G . The reason we write the restricted operator as D M is that its operation on the G-invariants agree with the operation of D on the sections of P × G E.
Note that D M on (C ∞ (P ) ⊗ E) G is effectively elliptic and Fredholm. Moreover, if we write the (scalar) Laplacian on P as ∆ P , then D 2 M + ∆ P is equal to an operator F of order zero (see [2,Prop. 5.6, p. 172]). So L := −∆ P + F is a generalized Laplacian that agrees with D 2 M on (C ∞ (P ) ⊗ E) G . We shall denote by P t the heat kernel associated with L, that is, the integral kernel of the operator e tL . For Here Str denotes the super trace for graded operators. Thus, the pairing of [D M ] with a function f ∈ C ∞ (G) can be calculated in the following way: Equation (6) shows that [D M ] indeed has point support at the identity, owing to the finite-propagation property of the heat kernel (see [5,Prop. 7.24, p. 107]).

Chern-Weil Forms
We quickly recall the construction of the Chern-Weil homomorphism. As a preliminary remark, suppose we have a formal power series ϕ ∈ R[[g * ]]. Let ∧(N ) be the exterior algebra generated by some finite-dimensional vector space N over R, and let ∧ + (N ) be its subalgebra comprising all elements of even degree. Then the formal power series ϕ defines a map g ⊗ ∧ + (N ) → ∧ + (N ) in the following way. Identify g with g⊗{1} ⊂ g⊗∧ + (N ). By duality, the evaluation of χ ∈ g * at an arbitrary element η = X j ⊗ η j ∈ g ⊗ ∧ + (N ) takes the value χ(η) = χ(X j )η j . The evaluation map ev η : g * → ∧ + (N ), χ → χ(η), extends uniquely as an algebra homomorphism to ev η : R[[g * ]] → ∧ + (N ). Then ev η (ϕ) =: ϕ(η) is the evaluation of ϕ at η. Note that ev η factors through S(g). All of this makes sense even if we replace N with N C := N ⊗ C. Now let Ω(P ) denote as usual the algebra of differential forms on P . The pullback κ * : Ω(M ) → Ω(P ) induced by the bundle projection is an injective algebra homomorphism; its image is the algebra Ω bas (P ) of basic forms on P . So κ * has a left-inverse (pushforward), which we denote by κ * : Ω bas (P ) → Ω(M ).
The curvature Θ of our connection θ is an element of g ⊗ Ω + bas (P ), so it makes sense to evaluate a formal power series ϕ ∈ R[[g * ]] at Θ/2πi ∈ g ⊗ Ω + bas (P ) C . The resultant ϕ(Θ/2πi) is a basic form on P , so we can apply the pushforward κ * . This process yields an algebra homomorphism, namely, We refer to CW(ϕ) as the Chern-Weil form of ϕ. The high point of Chern-Weil theory is that the de Rham cohomology class of the Chern-Weil form CW(ϕ) is a characteristic class. As demonstrated by Berline and Vergne [3], a similar "construction" occurs in heat kernel calculations. This is because there is a vector space isomorphism spin(n) ∼ = ∧ 2 (R n ) by virtue of the Chevalley map σ : Cl(n) → ∧(R n ), which is defined, in terms of the standard orthonormal basis {e i } n i=1 of R n , by the equation σ(e i 1 · · · e i k ) = e i 1 ∧ · · · ∧ e i k for any subset {e i 1 , . . . , e i k } of the basis. The Chevalley map is a vector space isomorphism, and the image of spin(n) is exactly ∧ 2 (R n ). The calculation that mimics the construction of the Chern-Weil map is captured in Lemma 4.2 below. But first, we setup some notations.
Definition 4.1. We denote by the composition of the Lie algebra homomorphism (5) with the Chevalley map. We set  has an asymptotic expansion ∞ k=0 Ψ k t k for t → 0+. (The asymptotic expansion is independent of the choice of ψ.) The kth coefficient Ψ k is contained in k q=0 ∧ 2q (R n ). If k ≤ n/2, then the component of Ψ k of degree 2k (the highest degree part) is equal to that of ϕ(Λ) ∈ ∧ + (R n ).
Remark on the proof. This lemma is similar in form to Lemma 11.3 in Duistermaat [4, p. 137]. The proof given there can be carried over almost verbatim. The only extra thing that needs to be checked is is an orthonormal basis for g; this can be verified using the Jacobi identity of the Lie bracket. We omit the details.
The element ϕ(Λ) appearing in Lemma 4.2 is a Chern-Weil form in disguise (provided that ϕ is G-invariant), as implied by the next lemma. Before stating the lemma, recall that we have a smooth map c : Cl(M ) → Cl(n) that is an algebra isomorphism when restricted to Cl(T x M ) ⊂ Cl(M ) for any x ∈ M . This yields, via the Chevalley identification, a smooth map ∧(T M ) → ∧(R n ) that is a vector space isomorphism when restricted to ∧(T x M ). Though it is an abuse of notation, we shall denote this map also as c : ∧ (T M ) → ∧(R n ).
One more notation; consider the map ♯ : Ω 1 (M ) → X(M ) that maps a 1form to its dual vector field relative to the metric. This induces an algebra isomorphism ♯ : Ω(M ) → ∧X(M ), which maps differential forms to polyvector fields (so-called the "raising of indices"). Lemma 4.3. Consider the polyvector field (κ * Θ) ♯ , namely, the one obtained by taking the pushforward κ * Θ of the curvature form Θ along the bundle projection and then raising its indices. Denote the value of this polyvector field at an arbitrary point x ∈ M by (κ * Θ) ♯ x . Then c((κ * Θ) ♯ x ) = Λ. Remark. A straightforward consequence is that, for any ϕ ∈ R[[g * ]] g and any x ∈ M , we have where A is the following composition of algebra homomorphisms: x M, where the first map is the evaluation at Λ/2πi; the second is the inverse of ∧(T x M ) c − → ∧(R n ); and the last is the lowering of indices.
Proof. For X ∈ g, let X, Λ denote the inner product of X with the gfactors of Λ, so that X, Λ = λ(X). We need to check that this is equal to c X, κ * Θ ♯ x . Since λ(X) ∈ ∧ 2 (R n ), we may write The curvature form Θ and the Riemannian connection ∇ P for X(P ) satisfy The right-hand side is, by definition, α x (X)ξ i , ξ j . This proves that c X, κ * Θ ♯ x = λ(X) as desired.

Proof of the Theorem
Theorem 5.1. Let M be a closed oriented Riemannian manifold of dimension n. Let P be a principal bundle over M whose fibers are isomorphic to a compact Lie group G. Let E be a G-vector space that is also a graded Cl(n)- for any G-invariant analytic function ϕ defined on some neighborhood of the origin in g. The pairing is zero if n is odd.
Proof. We begin by recalling the definition of the Duflo isomorphism in terms of distributions. Let E ′ (g) G 0 denote the algebra of G-invariant distributions on g supported at the origin. Likewise, let E ′ (G) G e denote the algebra of G-invariant distributions on G supported at the identity. Let j g be the analytic function on g defined by: j g (X) = det 1/2 sinh(ad(X/2)) ad(X/2) .
The Duflo isomorphism is then defined as where exp * denotes the pushforward induced by the exponential map, and j denotes the multiplication by j g .
Let exp * : C ∞ (G) → C ∞ (g) be the pullback along the exponential map. Since the exponential map is a local diffeomorphism on some neighborhood U of the origin in g, there is an isomorphism to which exp * serves as a left-inverse. This induces, by duality, a linear map , which is inverse to exp * . Then we have Duf −1 = j −1 • log * , so Here we have included a suitable G-invariant bump function ψ so that the pullback log * (j −1 g ψϕ) makes sense. This is fine since [D M ] has point support. Applying Equation (6), we get: Because the left-hand side is independent of t, it is sufficient to show that the right-hand side is asymptotically equal to D φ,M as t → 0+ when n is even, and that it is of O(t 1/2 ) when n is odd. We focus our attention to the integral over G, Let j M (X) := det 1/2 sinh(α(X/2)) α(X/2) .
And let vol denote the Riemannian volume form on M associated with the measure dx on M . We claim that, if n is odd, then for t → 0+; if n is even, then where CW is the Chern-Weil map induced by the connection θ on P (which is determined by the Clifford connection associated with D), and the decoration | top picks out the top degree part of the differential form at hand. Assume for the moment that the claim is true; then, since the de Rham cohomology class of the Chern-Weil form CW(j M e −τ ) is the index class of D, we indeed have Equation (8).
To prove our claim, we change the domain of the integral I(t) from G to g by means of the exponential map. As it is well-known, j 2 g (X) calculates the Jacobian determinant of the exponential map when the exponential map is diffeomorphic near X. So Str P t (x, x · exp(X))j g (X)ψ(exp(X))ϕ(X)e −ν * (X) dX.
In the limit t → 0+, the function t → P t (x, x · exp(X)) is of O(t ∞ ) if X is outside any neighborhood of the identity. Thus, the bump function ψ may be dropped without affecting the asymptotic behavior. In other words, Str P t (x, x · exp X)j g (X)ϕ(X)e −τ (X) e −γ(X) dX as t → 0+. Note that we have used the relation ν * = γ + τ .
The asymptotic expansion of P t is well-known. See, for instance, Berline, Getzler, and Vergne [2, Thm. 5.9, p. 176], from which we deduce that where Φ 0 (x, X) = j g (X)j −1 M (X)e −τ (X) ϕ(X). The rest of the argument proceeds similarly to that found in Berline and Vergne's proof [3] of the Atiyah-Singer index theorem (see [2, § 5.4]). In short, by Lemma 4.2 and the representation theory of Clifford algebras, we conclude that I(t) = O(t 1/2 ) if n is odd, and if n is even. Owing to Equation (7), we have when n is even. Hence Equation (9) and (10) hold as claimed, and we are done.