A proof of Duflo’s polynomial conjecture for nilpotent Lie groups

Let [Formula: see text] be a connected and simply connected real nilpotent Lie group with Lie algebra [Formula: see text], [Formula: see text] an analytic subgroup of [Formula: see text] with Lie algebra [Formula: see text], [Formula: see text] a unitary character of [Formula: see text] and [Formula: see text] the monomial representation of [Formula: see text] induced from [Formula: see text]. Let [Formula: see text] be the algebra of the [Formula: see text]-invariant differential operators on the line bundle over [Formula: see text] associated to the data [Formula: see text]. We know that [Formula: see text] is written as [Formula: see text], where [Formula: see text] with a certain [Formula: see text] verifying [Formula: see text]. Let [Formula: see text] be the symmetric algebra of [Formula: see text] and [Formula: see text] Then, [Formula: see text] possesses the Poisson structure [Formula: see text] well determined by the equality [Formula: see text] if [Formula: see text] are in [Formula: see text]. Let us consider the algebra [Formula: see text] of [Formula: see text]-invariant polynomial functions on the affine subspace [Formula: see text] of [Formula: see text]. This inherits the Poisson structure from [Formula: see text]. We denote by [Formula: see text] its Poisson center and [Formula: see text] the center of [Formula: see text]. We provide in this paper a positive solution to a problem due to Duflo ([8, Problem 3]), stating that the algebras [Formula: see text] and [Formula: see text] are isomorphic. Only some particular cases have been treated so far (cf. [14]).