My PhD research dealt mainly with various aspects of the category O. In connection with this I saw the beauty of the Kazhdan-Lusztig polynomials. Blocks of O along with some of its functors categorify the correspoding Hecke algebra (in which the Kazhdan-Lusztig polynomials originate), so a great deal can be learnt about Category O by studying the Hecke algebra. My recent focus has been on studying decompositions of the Hecke algebra as a module over itself, and what these can tell us about Category O.
The main focus of my thesis was various aspects of Kostant's Problem, which for a given module M asks wether a certain subring of the endomorphism ring of linear maps on M is isomorphic to a canonical subquotient of the enveloping algebra.