Criteria for the Finiteness of Restriction of U(g) - Modules to Subalgebras and Applications to Harish-Chandra Modules. A Study in Relation to the Associated Varieties

Let g be a finite-dimensional complex Lie algebra, and let U(g) be the enveloping algebra of g. Simple criteria are given for finitely generated U(g)-modules H to remain finite under the restriction to subalgebras A ⊂ U(g), by using the algebraic varieties in g* associated to H and A. It is shown that, besides the finiteness, the U(g)-modules H satisfying our criteria preserve the Gelfand-Kirillov dimension and Bernstein degree under the restriction to Lie subalgebras. Applying these results to Harish-Chandra modules of a semisimple Lie algebra g, we specify, among other things, a large class of Lie subalgebras of g on which all the Harish-Chandra modules are of finite type. This allows us to extend the finite multiplicity theorems for induced representations of a semisimple Lie group, given in earlier work [H. Yamashita, J. Math. Kyoto Univ.28 (1988), 173-211; J. Math. Kyoto Univ.28 (1988), 383-444].