Quantum Groups, the Loop Grassmannian, and the Springer Resolution

In this diagram, G is a connected complex semisimple group of adjoint type with Lie algebra g. We fix a Borel subgroup B ⊂ G, write b = Lie B ⊂ g for the corresponding Borel subalgebra, and n for the nilradical of b. Let Ñ := G×B n be the Springer resolution, and CohG×C ∗ (Ñ ) the abelian category ofG×C∗-equivariant coherent sheaves on Ñ , where the group G acts on Ñ by conjugation and C∗ acts by dilations along the fibers. Furthermore, let U be the quantized universal enveloping algebra of g specialized at a root of unity. The category block(U) on the left of (1.1.1) stands for a mixed version; see [BGS, Definition 4.3.1] or Section 9.2 below, of the abelian category of finite-dimensional U-modules in the linkage class of the trivial 1-dimensional module. Finally, we write DC for the bounded derived category of an abelian category C. Forgetting part of the structure one may consider, instead of block(U), the category block(U) of actual (nonmixed) U-modules as well. Forgetting the mixed structure on the left of diagram (1.1.1) corresponds to forgetting the C∗-equivariance in the middle term of (1.1.1), i.e., to replacing G × C∗-equivariant sheaves on Ñ by G-equivariant ones. Although this sort of simplification may look rather attractive, the resulting triangulated category D coherent(Ñ ) that will have to replace the middle term in the diagram above will no longer be the derived category of the corresponding abelian category Coh(Ñ ) and, in effect, of any abelian category. This subtlety is rather technical; the reader may ignore it at first reading. Finally, let G∨ denote the complex connected and simply-connected semisimple group dual to G in the sense of Langlands. We write Gr for the loop Grassmannian of G∨. The Grassmannian has a standard stratification by Iwahori (= affine Borel) orbits. The strata, usually called Schubert cells, are isomorphic to finitedimensional affine-linear spaces. We let Perv(Gr) denote the abelian category of