Grothendieck resolution, affine Grassmannian, and Yangian
In this thesis we address several questions on the structure and representation theory of quantum groups. The inspiration for problems and obtained solutions come from Poisson geometry. In the first chapter we classify the symplectic leaves on (a Poisson ind-subvariety of) the thick affine Grassmannian. Our classification extends previous results on the Poisson geometry of SL(n)-monopoles and the thin affine Grassmannian in type A. In the second chapter we show that a generic symplectic leaf (of maximal dimension) allows for a natural atlas and study transition maps between the charts. In particular we obtain a family of intertwining operators for the Yangian Y(gl(n)) by quantizing these transition maps. Finally, the third chapter is devoted to a different but closely related problem. Namely, we construct a quantum version of the multiplicative Grothendieck-Springer resolution. As a corollary we obtain an embedding of the quantum group into the algebra of “quantum differential operators” on the principal affine space, realized as the Heisenberg double of the quantum Borel subalgebra.