Research Interest
My current research interest is representations
of reductive Lie Groups. In particular, I am interested in representation
theory related to nilpotent
orbits of real and complex Lie algebras.
The study of nilpotent orbits partly
stems from the orbit method (or quantization, in a broader context), which
roughly means attaching a unitary representation to any (co)adjoint
orbits in a Lie algebra.
For classical Lie groups, i.e. GL(n,C), SO(n,C)
or Sp(2n,C) (or R), there is a nice combinatorial way of describing the
nilpotent orbits using tableaux.
This gives a nice bridge between combinatorics and
geometry. For instance, I am interested in the structures
of the diagram below:
For experts:
-
The numbers 1,2,3,4 are the
permutations of four elements. The elements above have higher Bruhat order than those
below.
-
The elements in the brackets are
the tau-invariants.
-
The linked graphs are the
left-cells.
-
The ' blocks ' are the left Young
tableaux obtained by Robinson-Schensted
algorithm. Note that every linked element has the same left tableau.
Amazingly, these objects, which are
interesting in themselves combinatorially, have deep
relationships with representation theory. More can be found in my short notes
on Kazhdan-Lusztig polynomials.
As you may have heard from
mainstream media, a
group of people computed something in E8 in 2007. To be more precise, they
computed the Kazhdan-Lusztig
polynomials (or more precisely, Kazhdan-Lusztig-Vogan
polynomials) for all real types of the exceptional Lie Group E8. The program
they are using is called atlas, which is available here.