Mathematical Sciences: Unipotent Representations for Reductive Groups

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The principal investigator plans to work on problems in mathematical analysis concerning the representation theory of Lie groups. A suitable example of the latter is the group of rotations of a sphere. Groups like this are important because they occur in many areas of mathematics ( e.g. geometry, differential equations,algebraic number theory, mathematical physics ) as groups of symmetries. Representation theory allows one to take advantage of symmetries in solving problems. The building blocks for representation theory are the irreducible representations, particularly the irreducible unitary representations. The principal investigator will work toward classifying these representations for certain classes of Lie groups. More specifically, he will continue work on determining the character theory and unitarity of the unipotent representations of reductive groups. This will provide significant insight into the unitary dual and certain problems in the theory of automorphic forms.