Applications de la bi-quantification à la théorie de Lie

This article in French, with a large English introduction, is a survey about applications of bi-quantization theory in Lie theory. We focus on a conjecture of M. Duflo. Most of the applications are coming from our article with Alberto Cattaneo [13] and some extensions are relating discussions with my student [9]. The end of the article is completely new. We prove that the conjecture E = 1 implies the Kashiwara-Vergne conjecture. Our deformation is non geometric but uses a polynomial deformation of the coefficients. Thanks : I thank the organizers of the conference “Higher Structures in Geometry and Physics” which held at IHP, January 2007 and where our results have been announced. This article is dedicated to Murray Gerstenhaber and Jim Stasheff. 1 English Introduction 1.1 Invariant differential operators on line bundle Let G be a real Lie group, connected and simply connected. Let g the associated Lie algebra, U(g) the universal enveloping algebra and S(g) the symmetric algebra. In this introduction G/H is a homogeneous space with H a connected Lie sub-group. As usual h denotes the Lie algebra of H . Fix a character λ of H , it’s a group homomorphism from H into C. If there are no danger of confusion, we will denote by the same letter the differential of the character. So λ is a character of h, ie. we have λ[h, h] = 0. Let’s Lλ be the line bundle defined by λ. Sections of this bundle, denoted by Γ (Lλ), are smooth functions on G such that φ(gh) = φ(g)λ(h). Obviously G acts on the left on Γ (Lλ). Let Dλ be the algebra of invariant differential operators on Γ (Lλ). After Koornwinder [24] we know Dλ is isomorphic to Dλ := ( U(g)C/U(g)C · h−λ )h (1) where h−λ = {H − λ(H)}. Here are some explanations. For X ∈ g, RX is the left invariant vectors field on G associated to X . For u ∈ Dλ, let Du be the associated differential operator on Γ (Lλ) defined by (Duφ)(g) = (Ruφ)(g), φ ∈ Γ (Lλ). Then Du in a left invariant differential operator. It’s not difficult to verify that we have described all of them. Suppose λ real. The algebra ( U(g)/U(g) · h−λ )h is not commutative in general. A conjecture of M. Duflo [15] describes the center of this algebra. I write Sλ the algebra of H-invariant polynomial functions on h ⊥ −λ := {f ∈ g, f |h = λ}. We get Sλ := ( S(g)/S(g) · h−λ )h . (2) This space admits a natural Poisson structure coming from the classical Poisson structure on g. Put δ(H) = 12 trg/h ad(H) the character for the half densities. Duflo’s conjecture [15] : The center of ( U(g)/U(g) ·h−λ−δ )h is isomorphic to the Poisson center of Sλ. This conjecture is far to be solved. Moreover one should probably ask for generic character. In case G is a nilpotent group, appreciable advances have been achieved in last few years by Corwin-Greenleaf [14], Fujiwara-Lion-Magneron-Mehdi [18], Baklouti-Fujiwara [7] and Baklouti-Ludwig [8] and Lipsman [27, 28, 29]. More precisely in the nilpotent case one can prove the following. Théorème 1 ([18]) Let G nilpotent (connected, simply connected) and χ the unitary character of H defined by χ(expG(H)) = exp(iλ(H)). 1 Consider the local expression around the origin. 2 Consider the following example ; G is reductive H = U the unipotent radical of a Borel. Let’s note t a Cartan subalgebra. Then you get (U(g)/U(g) · h) = S(t) = (S(g)/S(g) · h) . These algebras are commutative. The space G/U is quasi-affine.