Professor Enright's program of mathematical research has three main aspects. The first is the continuation of a study investigating the connections among systems of differential equations, certain affine varieties called determinantal varieties, and highest weight representations for a Lie group. The second objective is to present a theory of special functors on categories of modules for a Lie algebra which will allow the extension of the functorial methods of Zuckerman and Vogan to a domain which includes Kac - Moody Lie algebras. The third aspect is a study of unitarizability for representations of Kac -Moody algebras. 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. Transfer of this information, say between two different ways of keeping track of the representations of a single group or algebra, is something Professor Enright is involved in studying. Although Lie objects in the classical sense are finite-dimensional, there has lately been a great deal of interest in their infinite-dimensional analogues, whose investigation is also part of this project.
