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Deformation quantization and invariant distributions

In 1979, M. Kashiwara and M. Vergne formulated a conjecture on a Lie group G which implies that the Duflo isomorphism of Z(g) and S(g)^g extends to a natural module isomorphism between the spaces of germs of invariant distributions on G and g=Lie(G), respectively. They also proved their conjecture for G solvable. Using Kontsevich's deformation quantization we establish directly this result for distributions on any real Lie group G. In turn this gives a new proof of Duflo's result on the local solvability of bi-invariant differential operators on G.


Invariant distributions
Let G be a finite dimensional real Lie group with Lie algebra g. We write U = U(g), S = S(g) for the enveloping and symmetric algebras of g, and denote by I = S g , Z = U g the subalgebras of g-invariants.
Let U and S be the spaces of germs at 1 and 0 of distributions on G and g, and let Z and I be the subspaces of g-invariant distributions. We identify U and S with the convolution algebras of distributions with point support 1 ∈ G and 0 ∈ g. Then we have the following inclusions We denote by exp * : S −→ U and exp * : U −→ S the pushforward and pullback of distributions (germs) under the exponential map; and define η : S −→ U by where q is the following analytic function on g : Using the orbit method initiated by Kirillov, Duflo [Du] proved that the restriction of η to I is an algebra isomorphism from I to Z, i.e. η(p 1 * g p 2 ) = η(p 1 ) * G η(p 2 ) for p 1 , p 2 ∈ I where * g and * G denote convolution in the Lie algebra and Lie group respectively.
Let D be the algebra of germs at 0 of analytic coefficient differential operators on g, and let R be the right ideal of D generated by the germs of the adjoint vector fields By virtue of the pairing between functions and distributions, S is naturally a right D-module, and to emphasise this fact we shall write the action on distributions on the right.
We shall establish the following result which was conjectured by Kashiwara and Vergne, and proved by them for solvable groups [KV,Corollary 1].
Theorem 1. -For p in I, the following differential operator D p : S → S lies in R: As is explained in [KV], this result has several important consequences. First, since operators from R annihilate distributions from I, we obtain the following extension of the Duflo isomorphism : Theorem 2. -For p in I and P in I, we have η(P * g p) = η(P ) * G η(p).
Next, recall that a distribution Q on G is called an eigendistribution if there is a character χ : Z −→C such that Q * G z = χ(z)Q for all z in Z Eigendistributions on the Lie algebra are defined similarly, and we obtain the following: Theorem 3. -The map η takes germs of invariant eigendistributions on the Lie algebra to those on the Lie group.
Another consequence is the following important result, which was conjectured by L. Schwartz and proved by Duflo in [Du] by different methods. (See also [He], [Ra], [DR] for various special cases): Corollary. -Every nonzero bi-invariant differential operator on G is locally solvable. This follows immediately from Theorem 2 and Rais' lemma from [Ra] on the existence of invariant fundamental solutions on the Lie algebra g.

Deformation quantization
We will deduce Theorem 1 from careful analysis of M. Kontsevich's remarkable result on the formality of the Hochschild complex of a smooth manifold [Ko].
Kontsevich's main result leads to the construction of an explicit ⋆-product on the algebra of functions on an arbitrary smooth Poisson manifold (X, γ). This ⋆-product is given by a formal power series of bidifferential operators (depending on a parameterh).
In the special situation when X = g * is the dual of a finite dimensional Lie algebra g, and γ is the usual Poisson bracket (dual to the Lie bracket); we can seth = 1 and Kontsevich's ⋆-product descends to an actual product on the algebra of polynomial functions on g * , which can be naturally identified with S = S(g).
If, as before, we consider S as the algebra of distributions with point support (at 0 ∈ g), then the ⋆-product can be viewed as a new convolution on S. The advantage of the distributional point of view is the following key result: Theorem 4. -Given p ∈ S there is a unique element ∂ ⋆ p in D such that for all r ∈ S r ⋆ p = r · ∂ ⋆ p .
(As explained in the previous section, we are writing differential operators on the right.) The proof of this result involves examining the Kontsevich ⋆-product formula, replacing multiplications by differentiation and vice versa. This results in an expression for ∂ ⋆ p as an infinite sum of polynomial coefficient differential operators of bounded order. The coefficients of the sum are given by certain integrals which, while hard to compute, can be readily estimated to deduce Theorem 4. The linearity of γ is crucial in these considerations.
Next, in [Ko,8.3.3] Kontsevich introduces a certain formal power series where the constants c 2k (denoted c (1) 2k in [Ko]) are expressed as integrals of certain smooth differential forms ("angle forms") over certain compact manifolds with corners (compactified configuration spaces, introduced in [FM]). It can be checked that c n ∼ α n for some positive constant α. Hence in a neighborhood of 0 the power series S 1 (x) converges to an analytic function which we shall denote by τ (x).
Theorem 5. -For p ∈ I, the operator T p = ∂ p τ − τ ∂ ⋆ pτ belongs to R. Here ∂ ⋆ pτ is the differential operator defined in Theorem 4 and r · ∂ p := r * g p. To establish this result, we study the expression (r * g p)τ − (rτ ) ⋆ (pτ ) for r ∈ S and p ∈ I, as in section 8.2 of [Ko]. This enables us to express T p as an infinite sum of differential operators each belonging to R, and Theorem 5 follows from Theorem 4.
Theorem 1 is now a consequence of the following lemma: Lemma 1. -For p ∈ S, we have D p = T p τ −1 q.
To prove this lemma it is enough to show that r · D p = r · (T p τ −1 q) for r in S. In turn, this follows from the fact that there is a canonical isomorphism κ : S → U (denoted I −1 alg in [Ko]) which satisfies the relations [Ko,8.3.1] κ(r ⋆ p) = κ(r) * G κ(p) and [Ko,8.3.4] κ(rτ ) = η(r) = exp * (rq) for r ∈ S.