Sur lʹinverse de lʹapplication de Dixmier pour une algèbre de Lie résoluble

Résumé Let be a completely solvable Lie algebra over a field of characteristic zero. Let Â( ) be the algebra of differential operators with formal power series coefficients. The subalgebra generated by the Weyl algebra A( ) on and the formal power series with rational coefficients in the weights of the adjoint representation is denoted by ( ). There is a canonical imbedding L of the enveloping algebra U( ) in ( ). For every derivation x of , the adjoint vector field on defined by −x is denoted by W (x). Let a be the biggest nilpotent ideal in the algebraic hull of the image of by the adjoint representation. For every prime ideal I in U( ), let M (I) be the quotient of ( ) by the left ideal generated by L (I) and W ( a). The main result connects β (I) and M (I) where β (I) is the inverse image of the Dixmier map for . To state this result, certain categories ModJ[ ( )] are introduced and β (I) is described in terms of the set of all J such that M (I) is in ModJ[ ( )].