Similarity of blocks in parabolic category O and a wonderful correspondence for modules of covariants.

Let (g,k) be the pair of complexified Lie algebras corresponding to an irreducible Hermitian symmetric space of noncompact type. Then we have an associated triangular decomposition g = p− ⊕ k ⊕ p+, and q = k ⊕ p+ is a maximal parabolic subalgebra of g with abelian nilradical. Associated to the pair (g,q) we can define a highest weight category O(g,q) which is a parabolic analogue of the BGG category O. Each block in O(g,q) contains finitely many simple modules, whose highest weights form a nice partially ordered set. In this thesis a new notion, similarity of blocks, is introduced. Then, for the classical Hermitian symmetric pairs that are in the dual pair setting, a result is proved in each of the three classical cases that every block in O(g,q) is similar to some regular integral block. As an application of similarity, a foray into classical invariant theory leads to a “wonderful correspondence” between certain modules of covariants. This correspondence, which was previously introduced for a few special cases, is extended to all modules of covariants (in the classical cases) that are Cohen-Macaulay.