Quantum groups, the loop Grassmannian, and the Springer resolution

Abstract

We establish equivalences of the following three triangulated categories: \[ D quantum ( g ) ⟷ D coherent G ( N ~ ) ⟷ D perverse ( G r ) . D_\text {quantum}(\mathfrak {g})\enspace \longleftrightarrow \enspace D^G_\text {coherent}(\widetilde {{\mathcal N}})\enspace \longleftrightarrow \enspace D_\text {perverse}(\mathsf {Gr}). \] Here, D quantum ( g ) D_\text {quantum}(\mathfrak {g}) is the derived category of the principal block of finite-dimensional representations of the quantized enveloping algebra (at an odd root of unity) of a complex semisimple Lie algebra g \mathfrak {g} ; the category D coherent G ( N ~ ) D^G_\text {coherent}(\widetilde {{\mathcal N}}) is defined in terms of coherent sheaves on the cotangent bundle on the (finite-dimensional) flag manifold for G G ( = = semisimple group with Lie algebra g \mathfrak {g} ), and the category D perverse ( G r ) D_\text {perverse}({\mathsf {Gr}}) is the derived category of perverse sheaves on the Grassmannian G r {\mathsf {Gr}} associated with the loop group L G ∨ LG^\vee , where G ∨ G^\vee is the Langlands dual group, smooth along the Schubert stratification. The equivalence between D quantum ( g ) D_\text {quantum}(\mathfrak {g}) and D coherent G ( N ~ ) D^G_\text {coherent}(\widetilde {{\mathcal N}}) is an “enhancement” of the known expression (due to Ginzburg and Kumar) for quantum group cohomology in terms of nilpotent variety. The equivalence between D perverse ( G r ) D_\text {perverse}(\mathsf {Gr}) and D coherent G ( N ~ ) D^G_\text {coherent}(\widetilde {{\mathcal N}}) can be viewed as a “categorification” of the isomorphism between two completely different geometric realizations of the (fundamental polynomial representation of the) affine Hecke algebra that has played a key role in the proof of the Deligne-Langlands-Lusztig conjecture. One realization is in terms of locally constant functions on the flag manifold of a p p -adic reductive group, while the other is in terms of equivariant K K -theory of a complex (Steinberg) variety for the dual group. The composite of the two equivalences above yields an equivalence between abelian categories of quantum group representations and perverse sheaves. A similar equivalence at an even root of unity can be deduced, following the Lusztig program, from earlier deep results of Kazhdan-Lusztig and Kashiwara-Tanisaki. Our approach is independent of these results and is totally different (it does not rely on the representation theory of Kac-Moody algebras). It also gives way to proving Humphreys’ conjectures on tilting U q ( g ) U_q(\mathfrak {g}) -modules, as will be explained in a separate paper.