Mathematical Sciences: Complex Geometry and Representation Theory of Lie Groups

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The theory of Lie groups, named in honor of the Norwegian mathematician Sophus Lie, has been one of the major themes in twentieth century mathematics. As the mathematical vehicle for exploiting the symmetries inherent in a system, Lie theory has had a profound impact upon mathematics itself and theoretical physics, especially quantum mechanics and elementary particle physics. The abstract theory of representations of Lie groups provides a list of the minimal linear realizations -- or in technical jargon, the irreducible representations -- of the group. These form the building blocks for all such realizations. Often, Lie groups arise as motions of geometric objects, as for example, the orthogonal group is the group of distance preserving motions of the sphere. In order to study the fine structure of such representations, or of the group itself, it is necessary to obtain realizations of the representation that reflect the inherent geometric structure. This has been an area of fundamental research in noncommutative harmonic analysis for several decades. Professor Rossi is an expert in this interface of representation theory and geometry, especially complex geometry. For a large class of Lie groups, the most accessible set of irreducible representations is the holomorphic discrete series, discovered in the 1950's by Harish-Chandra. However, there are many more representations of this type -- called highest weight representations -- that only recently have been realized concretely. Some of these representations, said to lie in the analytic continuation, relate directly to elementary particle physics and also to the study of differential equations. Professor Rossi has been in the forefront of the discovery and investigation of such representations. Recently, he observed that these geometric contexts would allow considerable generalization. This point of view connects representation theory with analysis and geometry of several complex variables. In his present research, Professor Rossi intends to explore these constructions and make precise which representations arise on vector bundles of forms on invariant domains in Grassmanians, and as far as possible develop the analytic tools to make a complete study of the structure of these representations. For example, the representations which arise in this way are subrepresentations of the tensor product of holomorphic discrete series and adjoints, and they should be in the analytic continuation of non-holomorphic discrete series.