The Campbell-Hausdorff formula and invariant hyperfunctions

Abstract

Let G be a Lie group and g its Lie algebra. We denote by V the underlying vector space of g. There is a canonical isomorphism between the ring Z(g) of the biinvariant differential operators on G and the ring I(g) of the constant coefficient operators on V which are invariant by the adjoint action of G. When g is semi-simple, this is the "Harish-Chandra isomorphism"; for a general Lie algebra, this was established by Duflo I-4]. We shall prove here, that when G is solvable the Duflo isomorphism extends to an isomorphism 9 of the algebra of "local" invariant hyperfunctions under the group convolution and the algebra of invariant hyperfunctions on V under additive convolution (the exact result will be stated below). This gives a partial answer to a conjecture of Rais 1-12]. The existence of such an isomorphism 9 is of importance for the harmonic analysis on G, and for the study of the solvability of biinvariant operators on G (see [7]). It reflects and explains the "orbit method" ([8, 9]), i.e. the correspondence between orbits of G in V*, the dual vector space of V, and unitary irreducible representations of G: let T be an irreducible representation of G, then the infinitesimal character of T is a character of the ring Z(fl). Let t~ be an orbit in V*, the map p~(P)=P(f) (f~ (9) is a character of the ring I(g) (I(g) being identified with the ring of invariant polynomials on V*). The principle of the orbit method is to assign to a (good) orbit tV a representation T~ of G (or g), whose infinitesimal character corresponds to p~ via the isomorphism ~. This is the technique used by M. Duflo to construct the ring isomorphism ~. Furthermore let t~ be (when defined) the distribution on V which is the Fourier transform of the canonical measure on the orbit 0, then t~ is clearly an invariant positive eigendistribution of every operator P in I (g) of eigenvalue p~(P). Kirillov conjectured that the global character of the representation T~ (when