Ho Hon Leung

Ph.D. (2011) Cornell University

First Position

Assistant professor at the Canadian University of Dubai

Dissertation

K-Theory of Weight Varieties and Divided Difference Operators in Equivariant KK-Theory

Advisor:

Reyer Sjamaar

Research Area:

symplectic geometry

Abstract: This thesis consists of two parts.

In the first part, we give a new proof of the K-theoretic analogue of Kirwan surjectivity theorem in symplectic geometry by using an equivariant version of Kirwan map. We also compute the kernel of this equivariant Kirwan map. As an application, we find the presentation of the K-theory of weight varieties, which are the symplectic quotients of complete flag varieties.

Demazure defined a set of isobaric divided difference operators on the representation ring of compact torus. It can be seen as a decomposition of the classical Weyl character formula. In the second part, we explicitly define these operators in the setting of equivariant KK-theory first introduced by Kasparov. It is a generalization of the results done by Demazure, Harada, Landweber and Sjamaar. Due to the elegance and generality of KK-theory, some interesting applications of the result are also given.