Let 9 be a complex semisimple Lie algebra. In ([8], Prop. 12) Duflo gave a remarkable sum formula interrelating induced ideals. The main result of this paper provides a natural generalization of this formula and more precisely gives a resolution for certain primitive quotients of the enveloping algebra U (g). The proof has three distinct steps.

One, the extension of the Bernstein-Gelfand-Gelfand (in short, BGG) resolution of a simple finite dimensional U (g) module to certain simple highest weight modules. Two, the description of the so-called t-finite part of the space of homomorphisms of any one Verma module to any other. Three, the proof of exactness of a certain functor. The last can be viewed as a non-trivial generalization of the fact that a Verma module with dominant highest weight is projective in the so-called C category. A by-product gives some results on a problem of Kostant relating U (g) to the t-finite part of the space of endomorphisms