Representation Theory of Reductive Groups

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Abstract Sahi The research will consist of two parts. The first part is concerned with constructing explicit models for small unipotent representations of reductive Lie groups, and with using these models to study the decomposition of tensor products and restrictions of such representations. It is expected that this will provide a natural context for extending the theta-correspondence. The second part of the proposal gives an extension of the theory of Jack polynomials, which are themselves generalizations of spherical polynomials for GL(n) where one replaces the root multiplicity by an arbitrary parameter. The polynomials we describe are obtained by varying the half-sum of positive roots, hence depend on n auxiliary parameters. It is expected that a study of these polynomials will lead to new and interesting results. The first part of the proposal will have applications to Number Theory, a deep and beautiful area of mathematics that has often had unexpected and profound implications for practical problems of various kinds. For example in the design of networks, in order to facilitate efficient dissemination of information, it is desirable to have a high degree of inter-connectivity. It turns out that some of the best networks are constructed by using number-theoretic considerations. The second part of the proposal has relevance to diverse areas such as Physics, Multivariate Statistics, and Combinatorics.