Quantum groups, the loop Grassmannian, and the Springer resolution

We establish equivalences of the following three triangulated categories:

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.