Classical Harish-Chandra modules appear from unitary representations of real complex semisimple groups. If G is a complex semisimple Lie algebra then a g-module M is a Harish-Chandra module if M is a direct sum of finite-dimensional modules over certain subalgebra k. This k in classical situation is the set of fixed point of a certain involution on g.

One can consider a more general situation when k is an arbitrary subalgebra in g. The well studied case is the case when the corresponding group K acts with finitely many orbits on the flag manifold. Beilinson-Bernstein classification of such modules is obtained from localization theorem. The Bernstein-Beilinson localization theorem is a very powerful method of solving many
problems in representation theory.

The other case when K is a maximal torus in G was studied recently in works of Fernando, Mathieu, Penkov, Dimitrov, Futorny and others by purely algebraic methods.

We will discuss how Beilinson-Bernstein construction can help in the case of general Harish-Chandra modules and prove some results for subalgebras containing a maximal torus.