Professors Corwin and Goodman will pursue a wide-ranging program of research in the representation theory of Lie groups and related objects. Part of Corwin's work will deal with invariant differential operators on homogeneous spaces of nilpotent groups. He will also study supercuspidal representations of p-adic groups. Goodman will continue investigating the representations of current algebras and loop groups. The algebraic properties of higher-order Sugawara operators will be studied using the theory of tensor invariants for the classical groups. Analytic aspects of the geometric models for positive-energy representations of loop groups will be examined. Goodman will also seek new types of Whittaker models for representations of real reductive groups. This project has to do with the representation theory of Lie groups (and of their cousins, Lie algebras), which bear the name of the Norwegian mathematician Sophus Lie. One basic example of a Lie group is the group of rotations of a sphere, where the group operation consists of following one motion by another. Detailed information concerning this group is very helpful in solving mathematical or physical problems in which spherical symmetry is present. Other groups of motions capture other kinds of symmetry. A more algebraic (as opposed to geometric) source of examples of Lie groups comes from the multiplication of matrices. The group of all invertible real (or complex) matrices of a given size is a Lie group, as is just about any subgroup thereof that can be described in a natural manner. It is desirable to be able to go back and forth between the geometric and algebraic points of view, for instance to consider the numerous ways in which the rotation group of the sphere can be realized as a group of invertible matrices. This, roughly, is representation theory. Facts about the representations of a given group tend to store a lot of information very economically. Depending on where the group in question comes from, this information can impinge on almost any subject from number theory to mathematical physics. The project of Professors Corwin and Goodman spans a fair portion of the applications and emphases of contemporary representation theory
