Abstract Sepanski Let G be a real reductive Lie group with K a maximally compact subgroup. Write g and k for the complexified Lie algebras, respectively, and let X be a (g, K) module. The objective of the investigator's research is to determine a non-combinatorial description for the set of K-types of X first in the case where X is a discrete series representation and second in the case where X is a cohomologically induced module. This will be accomplished in three steps: 1) make precise the notion of an "edge" of the set of K-types; 2) identify the K-types that lie on an edge using cohomological methods; and 3) use the convex hull of the edge K-types to describe the entire set of K-types. The investigator's methods are based on a conjecture of D. Vogan concerning a certain restriction map of cohomology. If q=l+u is a theta-stable parabolic with Levi component l, Vogan's conjecture states that the image of the restriction map from u cohomology to u intersect k cohomology parameterizes the K-types lying on edges that are not contained in a Weyl chamber wall of K. The ideas associated to this conjecture provide the main techniques for approaching the proposed objective. From a global point of view, this project is motived by the existence of symmetry in the world. This symmetry manifests itself in a variety of forms such as the lattice structure of certain crystals in chemistry, the behavior of subatomic particles in high energy physics, or the equations describing fluid flow in engineering. In all these diverse applications, one of the mathematical tools that is particularly powerful is known as representation theory. In a broad sense, representation theory is the study of all possible symmetries of a given system. The hope is to classify all symmetries and to provide detailed information about each one. While great strides have been made towards this goal, many difficult problems remain. An important technique used to study these remaining problems is to examine somethin g known as the K-types of a representation. In a broad sense, the set of K-types is a slight simplification of a complicated representation to something that is more manageable. It turns out that the set of K-types is a very effective way to both construct representations and to deduce special properties of representations. The investigator's research aims to provide a geometrical description of the set of K-types that would make a more detailed analysis of representations possible.
