Critères d'induction et de coinduction pour certains anneaux d'opérateurs différentiels

Abstract

Let L be a Lie algebra acting by derivations on a commutative noetherian local ring ( R, M , K= R/ M ) and let V be the ring of differential operators built on R and L . Defining L ( M )={d∈ L /d( M )⊂ M } et L 0 = L ( M )/ M L : L 0 is a Lie algebra which acts on K by derivations, and we can construct a differential operators ring on K with L 0, denoted by V 0. With the help of the ( V 0 − V)-bimodule V/ M V we define the induction (resp. coinduction) from V 0 to V by Ind V v 0 =−⊗ v 0 V/ M V and we give a criterion for a V-module to be induced (resp. coinduced) from V 0. These results are similar to those established by Mackey for Lie groups and Blattner for Lie algebras, which are based on the notion of the system of imprimitivity.