## A geometric approach to the study of (g, k)-modules of finite type

- Let g be a semisimple Lie algebra over C and k be a reductive in g subalgebra. We say that a simple g-module M is a (g; k)-module if as a k-module M is a direct sum of finite-dimensional k-modules. We say that a simple (g; k) module M is of finite type if all k-isotypic components of M are finite-dimensional. To a simple g-module M one assigns interesting invariants V(M), V(M) and L(M) reflecting the 'directions of growth of M'. In this work we prove that, for a given pair (g; k), the set of possible such invariants is finite. Let K be a reductive Lie group with Lie algebra k. We say that a K-variety X is K-spherical if X has an open orbit of a Borel subgroup of K. Let W be a finite-dimensional K-module. The set of flags (W_1,..., W_s) of W with fixed dimensions (n_1;...; ns) is a homogeneous space of the group GL(W). We call such a variety partial W-flag variety. In this work we classify all K-spherical partial W-flag varieties. We say that a simple (g; k)-module is bounded if there exists constant C_M such that, for any simple k-module E, the isotypic component of E in M is a direct sum of not more than C_M-copies of E. To any simple sl(W)-module one assigns a partial W-flag variety. In this thesis we prove that a simple (sl(W); k)-module is bounded if and only if the corresponding partial W-flag variety is K-spherical. Moreover, we prove that the pair (sl(W); k) admits an infinite-dimensional simple bounded module if and only if P(W) is a K-spherical variety. For four particular case we say more about category of bounded modules and the set of simple bounded modules.