Embeddings of flag manifolds and cohomological components of modules
Tsanov, Valdemar Vasilev
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This thesis is a study in Representation Theory and Geometry. These two branches of mathematics have a fruitful interaction, with many applications to Physics and other sciences. Central objects of interest are homogeneous spaces and their symmetry groups. The geometric and analytic properties of homogeneous spaces relate to the structure and representation theory of the corresponding groups. The focus of this work is on certain representation theoretic phenomena related to equivariant embeddings of homogeneous spaces. The Borel-Weil-Bott theorem, a milestone in Representation Theory and Geometry, provides realizations for every irreducible module of a semisimple complex Lie group as various cohomology spaces of homogeneous vector bundles on flag manifolds of the group. The purpose of this dissertation is to initiate a study of the behaviour of the Borel-Weil-Bott construction under pullbacks along equivariant embeddings of flag manifolds. These pullbacks provide certain geometric branching rules for representations. This is where the notion of a cohomological component arises from. The central result of the dissertation is a criterion for nonvanishing of the pullback. The framework used for the formulation and proof of the result is Kostant's theory of Lie algebra cohomology. After the general criterion is established, various specializations and applications are presented: special classes of embeddings are considered, for which the criterion takes simpler forms; relations are established between pullbacks along embeddings of complete and partial flag manifolds; properties of the set of cohomological components are obtained; various examples are considered, the most interesting of which is related to the theory of invariants of semisimple Lie algebras.