Mathematical Sciences: Analysis on Semisimple Groups

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The principal investigator plans to work on problems in mathematical analysis concerning the representation theory of (semisimple) 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 semisimple Lie groups. Prior work provides a considerable store of examples and techniques for attacking this problem, which in the indicated degree of generality is very hard, and absolutely central to the subject. Some common themes have emerged and some subtle critical cases have appeared. The plan is to pursue the common themes and concentrate on the role played by generalizations of the critical cases.