While at the University of Oregon, some of my colleagues viewed me as an analyst. At the University of Washington, many of my colleagues label me an algebraist. How can this be? What the heck do I do anyway?
My research emphasis is in the area representation theory. The study of representations is directly linked to harmonic analysis of functions on the underlying Lie group. For example, the classical treatment of Fourier analysis on the unit circle T depends on identifying the dual group, which parametrizes the irreducible unitary representations of T. More generally, for a group like Sl(2,R), an explicit Plancherel formula depends upon some understanding of irreducible unitary representations. Now the problem is much harder, since the group Sl(2,R) is highly non-commutative. However, the group Sl(2,R) is a Lie group and we can effectively exploit the interaction of the group structure and differential structure to analyze various function spaces. (The boxed formula at the top of the page is the explicit Plancherel formula for Sl(2,R).)
Since every Lie group is the semidirect product of a reductive group and a solvable group, there is a natural fork in the road when studying the associated representation theory and harmonic analysis. My work has focused on the setting of a reductive Lie group. An expository account of the subject can be found by clicking here.
I work in four mainstream directions. I will highlight each below and you can link for more information if you wish. In addition, you will find out what these tiny compressed pictures are all about!
The structure and properties of Lie group representations.
Highest weight
representation theory.
Asymptotics of representations.
Singularity theory of representations.
You can link to my vita to check out my various research publications. Most of these are research articles or memoirs, but there are two books: Representations of rank one Lie groups, 1985 and Nilpotent orbits in semisimple Lie algebras, 1993 (which I co-authored with Monty McGovern).
