Unitary Dual of Real Groups

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AbstractSalamanca-Riba Dr. Salamanca Riba intends to investigate various problems in the representation theory of Lie groups. The first problem is the question of classifying all genuine unitary representations of the metaplectic group. Jeff Adams and Dan Barbasch have obtained a correspondence between these representations and a set of unitary representations of certain inner forms of the metaplectic group. The PI plans to study this correspondence and see if these representations go to interesting Zuckerman functor modules under this map. In addition, the PI will also work on a program already in progress in collaboration with David Vogan. The problem they want to solve is the following conjecture: There is a bijection between the set of unitary representations of a Lie group G, whose lowest K type is associated to a fixed parameter, and the set of unitary representations of a special subgroup with lowest K types associated to the same parameter.Lie groups have connections in many areas of applied Mathematics like Materials Science, Quantum Field Theory, Particle Physics, Control Theory and Robotics and Biology. For example, Control Theory and Robotics is related to some representations of certain compact Lie groups. In addition, Lie groups and their representations have also connections with other areas of pure Mathematics as well, like Differential Equations, Harmonic Analysis, Topology, Geometry and Ergodic Theory. For example, the study of problems in Analysis such as the behavior of solutions of differential equations, partial differential equations, integral equations, etc. leads naturally to formulating these problems not only in Euclidean space, but on differentiable manifolds. Many examples of these manifolds are classical Lie groups of matrices. This is particularly true of problems of this type arising in classical mechanics and physics and in modern problems of Lie groups and homogeneous spaces. Modern Analysis, when it goes beyond local results, becomes analysis on differentiable manifolds and Lie groups.,