Professor - Director of Graduate Studies
B.S., Haverford College, 1984
Ph.D., Massachusettes Institute of Technology, 1988
Algebra and Algebraic Geometry
Representation theory of Lie groups and geometry, including Poisson geometry and perverse sheaves. In particular, I study the interrelation between geometry and problems in the representation theory of Lie groups.
I am working on geometric aspects of representation theory. Using ideas of Beilinson, Bernstein, Lusztig, Springer, and others, one can construct many interesting representations using geometrical techniques. One then wants to relate properties of geometry to properties of representation theory. I am currently attempting to understand some questions about branching rules for infinite dimensional representations of real groups using these kinds of ideas. In addition, I am trying to understand microlocal invariants associated to representations.
In my paper with Jiang—Hua Lu on Poisson harmonic forms, we used ideas from Poisson geometry to better understand an interpretation of the cohomology ring of the flag variety using Lie algebra cohomology. This paper used a deformation argument, and in our paper on Lagrangian subalgebras, we study the space in which this deformation occurs. In this paper and in work in progress, we find Poisson structures on conjugacy classes in groups and on symmetric spaces which should be of interest. We would like to use structures of this nature to understand certain integral formulas.
- (with I. Mirkovic) Fourier transform and the Iwahori-Matsumoto involution, Duke Math. Journal (86)1997, pp.435—464
- On Springer representations and the Zuckerman functor, Pacific Journal of Mathematics(180)1997, pp.221—228
- (with I. Mirkovic) Characteristic cycles for the loop Grassmannian and nilpotent orbits, Duke Math. Journal (97)1999, pp.109—126
- (with J.-H. Lu) Poisson harmonic forms, Kostant harmonic forms, and the $S\sp 1$-equivariant cohomology of $K/T$, Advances in Math.(142)1999, pp.171—220
- (with J.-H. Lu) On the variety of Lagrangian subalgebras, Annales Ecole Normale Superieure,(34)2001, pp.631—668