On the existence of , -modules of finite type

Let be a reductive Lie algebra over an algebraically closed field of characteristic zero, and let be a subalgebra reductive in . We prove that admits an irreducible (,)-module M which has finite -multiplicities and which is not a (,′)-module for any proper inclusion of reductive subalgebras ⊂′⊂ if and only if contains its centralizer in . The main point of the proof is a geometric construction of (,)-modules which is analogous to cohomological induction. For = (n) we show that whenever contains its centralizer, there is an irreducible (,)-module M of finite type over such that coincides with the subalgebra of all g∈ which act locally finitely on M. Finally, for a root subalgebra ⊂ (n), we describe all possibilities for the subalgebra ⊃ of all elements acting locally finitely on some M.