Mathematical Sciences: Unitary Representations and Zuckerman Modules

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Abstract Salamanca-Riba During the duration of this grant Dr. Salamanca-Riba intends to investigate some extensions to her proof of the following conjecture of Vogan and Zuckerman on the representation theory of Lie groups: 'An irreducible unitary Harish-Chandra module of a group G whose infinitesimal character satisfies some regularity assumption is isomorphic to a Zuckerman functor module derived from a one dimensional unitary representation of a subgroup of G'. Relaxing this regularity assumption to include all genuine unitary representations of the metaplectic group, the PI plans to use the Adams-Barbasch dual pair correspondence of these representations to realize them as Zuckerman modules that are induced from some other representations easy to describe. The PI also has 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 G whose lowest K type is associated to a fixed parameter and the set of unitary representations of a certain subgroup associated to the same parameter and whose K types are in turn associated to that same parameter. A fourth problem she wants to study is those unitary representations of a Lie group G which are predicted by the orbit method. These representations, when restricted to the maximal compact subgroup K are multiplicity free and are related to multiplicity free representations on the ring of regular functions of a nilpotent orbit. Lie groups have connections in many areas of pure and applied Mathematics like Differential Equations, Harmonic Analysis, Topology, Geometry, Ergodic Theory; and in other areas of Science and Engineering as well, like Materials Science, Quantum Field Theory, Particle Physics, Control Theory and Robotics. For example, the PI has become interested in the applications of Lie Theory to the field of Geometric Control Theory. She is now exploring some mechanical systems that are interesting for applications in several very different areas, such as robotic assisted surgery, waste management and spacecraft control. These systems seem to have some common grounds that allow engineers to be able to control the systems and could possibly be explained with Lie Theory. She is also interested in exploring these questions a bit further to see if they lead to interesting mathematical questions and if the answers to those questions lead to practical applications in industry.