Research Statement

My interest centers in algebraic groups, Lie algebras and their representations from the perspective of algebra and geometry. More precisely, I am interested in the following topics:
Orbit structures in irreducible representations. The adjoint representations of semi-simple algebraic groups on their Lie algebras. Parabolic subgroups and parabolic subalgebras. Structure of stabilizer subgroups of elements of representations. Secant varieties of orbital varieties. Tensor products of irreducible representations. Spinor varieties. Categorification of representations.

Some of the questions I am working on are motivated by problems arising in my earlier research projects and by the results I have already obtained. Some of the questions have arisen during discussions with collaborators and during workshops and conferences. I describe two problem areas in the following.

• If V is the adjoint representation of a simple algebraic group on its Lie algebra we study the structure of the orbits in V. Specifically, we consider the orbits of parabolic subgroups on their nilradicals: How can we use the existence of a dense orbit in this setting? Or more generally, given a subset of an irreducible representation, what can we say about its support? What is the combinatorial description of this support?
• There are several ways of associating a category to an irreducible representation of (the universal enveloping algebra of) a Lie algebra. What is the dictionary between the properties of the representation and the properties of the category and of the associated functors? How do well-known results from representation theory translate to the categorical setting?