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

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## Abstract

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.