Mathematics Colloquia and Seminars

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Gieseker varieties arise as the Hamiltonian reduction of the space of framed representations of the Jordan quiver. Examples of these include closures of minimal nilpotent orbits in $\mathfrak{sl}_r$, as well as symmetric products of copies of $\mathbb{C}^2$.

Quantizations of Gieseker varieties may be defined via quantum Hamiltonian reduction, and they have an interesting representation theory. I will explain how to construct their representations using equivariant D-modules on $\mathfrak{sl}_r$ and how to study some of these representations by introducing an auxiliary algebra that is similar to the type A rational Cherednik algebra. This allows us to obtain character formulas for these representations. In particular, we obtain dimension and character formulas for finite-dimensional representations, as well as a combinatorial interpretation of these. This is based on joint work with Pavel Etingof, Vasily Krylov and Ivan Losev.