Quantum groups, the loop Grassmannian, and the Springer resolution

Abstract:

We establish equivalences of the following three triangulated categories: \[ D_\text {quantum}(\mathfrak {g})\enspace \longleftrightarrow \enspace D^G_\text {coherent}(\widetilde {{\mathcal N}})\enspace \longleftrightarrow \enspace D_\text {perverse}(\mathsf {Gr}).\] Here, $D_\text {quantum}(\mathfrak {g})$ 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 $\mathfrak {g}$; the category $D^G_\text {coherent}(\widetilde {{\mathcal N}})$ is defined in terms of coherent sheaves on the cotangent bundle on the (finite-dimensional) flag manifold for $G$ ($=$ semisimple group with Lie algebra $\mathfrak {g}$), and the category $D_\text {perverse}({\mathsf {Gr}})$ is the derived category of perverse sheaves on the Grassmannian ${\mathsf {Gr}}$ associated with the loop group $LG^\vee$, where $G^\vee$ is the Langlands dual group, smooth along the Schubert stratification. The equivalence between $D_\text {quantum}(\mathfrak {g})$ and $D^G_\text {coherent}(\widetilde {{\mathcal N}})$ is an “enhancement” of the known expression (due to Ginzburg and Kumar) for quantum group cohomology in terms of nilpotent variety. The equivalence between $D_\text {perverse}(\mathsf {Gr})$ and $D^G_\text {coherent}(\widetilde {{\mathcal N}})$ 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 $p$-adic reductive group, while the other is in terms of equivariant $K$-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 $U_q(\mathfrak {g})$-modules, as will be explained in a separate paper.