On the Determination of Irreducible Modules by Restriction to a Subalgebra

Let <!-- MATH $\mathcal{B}$ --> be an algebra over a field, <!-- MATH $\mathcal{A}$ --> a subalgebra of <!-- MATH $\mathcal{B}$ --> , and an equivalence class of finite dimensional irreducible <!-- MATH $\mathcal{A}$ --> -modules. Under certain restrictions, bijections are established between the set of equivalence classes of irreducible <!-- MATH $\mathcal{B}$ --> -modules containing a nonzero -primary <!-- MATH $\mathcal{A}$ --> -submodule, and the sets of equivalence classes of all irreducible modules of certain canonically constructed algebras. Related results had been obtained by Harish-Chandra and R. Godement in special cases. The general methods and results appear to be useful in the representation theory of semisimple Lie groups.