| In this thesis we study invariants of Harish-Chandra modules, for particular classes of groups G, namely Sp( p, q) and U(p, q). In Representation Theory, algebraic and combinatorial invariants are a powerful way to classify representations. Our interest is in the following question of Penkov and Zuckerman: What modules are admissible (in a precise sense) over a given ideal of k ? Here k is the symmetric subalgebra which is the complexification of the Lie algebra of the maximal compact subgroup of G. The approach we take is that of studying associated varieties (as defined by Vogan) which are particularly amenable to analyze, via combinatorics. This is so since nilpotent orbits defining associated varieties are parametrized by signed Young diagrams, which are rather concrete combinatorial objects. We achieve a full classification of the associated varieties of the admissible modules by studying the poset of signed Young diagrams and establishing new structure theorems. |
