This is a two-part talk. The second part is available here.
Symplectic singularities are a generalization of symplectic manifolds that have a symplectic form on the smooth locus but allow for certain well-behaved singularities. They have a strong relationship to representation theory and include nilpotent cones of semisimple Lie algebras, quiver varieties, affine Grassmannian slices, and Kleinian singularities. There is a combinatorial description for partial resolutions of conical affine symplectic singularities, stemming from Namikawa’s 2013 result that a symplectic resolution is also a relative Mori Dream Space. In this talk we will explore these partial resolutions in more detail, exploring their birational geometry, deformation theory, and Springer theory. In particular, we will review the definition of the Namikawa Weyl group for conical affine symplectic singularities and use birational geometry to define a generalization for their partial resolutions. We will also use this Namikawa Weyl group to classify the Poisson deformations of the partial resolutions. We will then describe how these partial resolutions fit into the framework of Springer Theory for symplectic singularities, following Kevin McGerty and Tom Nevins’ recent paper, Springer Theory for Symplectic Galois Groups. Finally, we will discuss some ongoing research that stems from these ideas, inspired by parabolic induction and restriction.
This video is part of the University of Georgia‘s Algebra seminar.
