A quantized Gieseker variety is an associative algebra quantizing the global
functions on a Gieseker moduli space. This algebra arises as a quantum Hamiltonian
reduction of the algebra of differential operators on a suitable space. It depends on
one complex parameter and has interesting and beautiful representation theory.
For example, when it has finite dimensional representations, there is a unique
simple one and all finite dimensional representations are completely reducible.
In fact, this is a part of an ongoing project with Pavel Etingof and Vasily Krylov,
one can explicitly construct the irreducible finite dimensional representation using
a cuspidal equivariant D-module on sl_n and get an explicit dimension (and character)
formula. This formula gives a “higher rank” version of rational Catalan numbers.
I’ll introduce all necessary definitions, describe the resuls mentioned above and,
time permitting talk about open problems.