ON THE STRUCTURE OF PRODUCED AND INIDUCED INDECOMPOSABLE LIE MODULES

l. Introd,ucti,on In recent years, it has become evident that certain indecomposable mod.ules are of basic importance to the representation theory of semisimple Lie algebras both over fields of characteristic 0 and over modular fields. If g is a semisimple Lie algebra and b a Borel sub-algebra of g, then these fundamental g-modules are those induced from one-dimensional b-modules (cf. [I2],[f],151,[a]). Also modules induced from more general parabolic subalgebras have recently revealed. some deep, far-reaching properties in the characteristic 0 setting (Lepowsky [7]). On the other hand, Humphreys ' work [4] indicates that a better understanding of induced indecomposable modules would be desirable in the modular case too. In this paper we study (restricted) indecomposable modules produced and induced from parabolic subalgebras of classical Lie algebras over fields of prime characteristic. We emphasize that,,produced " is being used here in the sense of [8],[13],[14]. Section 2 summarizes the basic definition