Invariant Theory and Transformation Groups

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The investigator studies the module theory of the algebra of invariant differential operators on a simple complex Lie algebra, in particular, he studies finite dimensional modules. He is currently considering the case of rank two Lie algebras, with success in the case of the Lie algebra of type A. The investigator and a colleague study commuting involutions on reductive algebraic groups, and the corresponding double coset spaces by the fixed groups of the involutions. We determine the structure of the double coset spaces, and their relations to quotient spaces of special kinds of split tori by appropriate Weyl groups.Symmetry groups, such as the collection of all rotations of a sphere, play a fundamental role in mathematics and physics. Understanding these groups and the ways they can act is basic to our knowledge of the physical and mathematical universes. Research on symmetry groups has been very active and productive for the last 75 years. The investigator and colleagues study these symmetry groups, related mathematical objects, and the different ways they can act and interact.