Quantum groups, the loop Grassmannian, and the Springer resolution

By Sergey V. Arkhipov, R. Bezrukavnikov, V. Ginzburg. 2003

146 citations recorded by the source corpus.

We establish equivalences of derived categories of the following 3 categories: (1) Principal block of representations of the quantum at a root of 1; (2) G-equivariant coherent sheaves on the Springer resolution; (3) Perverse sheaves on the loop Grassmannian for the Langlands dual group. The equivalence (1)-(2) is an `enhancement' of the known expression for quantum group cohomology in terms of nilpotent variety, due to Ginzburg-Kumar. The equivalence (2)-(3) is a step towards resolving an old mystery surrounding the existense of two completely different realizations of the affine Hecke algebra which have 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. Our equivalence (2)-(3) may be viewed as a `categorification' of the isomorphism between the corresponding two geometric realizations of the fundamental polynomial representation of the affine Hecke algebra. 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 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 representation theory of Kac-Moody algebras). It also gives way to proving Humphreys' conjectures on tilting U_q(g)-modules, as will be explained in a separate paper.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/62e6a5b70cdaafcbf1f780d4d788a90af3c45ca4

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Citations in the corpus

Sheaves on affine Schubert varieties, modular representations and Lusztig's conjecture (2007, unknown): “n by combining results of Kashiwara-Tanisaki, Kazhdan-Lusztig and Andersen-Jantzen-Soergel which form the three main steps in the Lusztig program (cf. [And95] for an overview), and, more recently, in [ABG04]. We give a new proof that is not invoking the Lusztig program in its whole, but uses only th”
On two geometric realizations of an affine Hecke algebra (2012, background): “ide in (30) is the derived category of I0 equivariant sheaves on the aﬃne Grassmannian, while the right hand is DGCohGˇ(˜gˇ L × gˇ{0}). This special case of (30) amounts to one of the main results of [2]; again one needs to apply linear Koszul duality [39] to pass from the coherent side of the equiv”
Tilting modules and the p-canonical basis (2015, methodology): “torial shadow” of the quantum analogue of the Finkelberg– Mirkovi´c conjecture [FM] which relates Rep0pGq to perverse sheaves on Gr. (This quantum version was proved by Arkhipov–Bezukavnikov–Ginzburg [ABG].) In the modular case, it can be easily deduced from the results presented in §1.6 that the po”
Schubert calculus and torsion explosion (2013, methodology): “rgel [AJS94], Kashiwara and Tanisaki [KT95, KT96], Kazhdan and Lusztig [KL93, KL94a, KL94b] and Lusztig [Lus94]. Alternative proofs for large p have been given by Arkhipov, Bezrukvanikov and Ginzburg [BG04], Bezrukvanikov, Mirkovic and Rumynin [BMR08] and Fiebig [Fie11]. Fiebig also gives an explici”
Perverse sheaves on affine flags and langlands dual group (2002, methodology): “In fact, this conjecture can be derived from a combination of the results of this paper and those of [ ABG ], or by adapting the method of [BGS]; see also discussion in [B4, 1.2].”
Koszul duality for Kac–Moody groups and characters of tilting modules (2017, methodology): “gory of ﬁnite-dimensional representations of Lusztig’s quantum group Uζ associated with an adjoint semisimple complex algebraic group G, specialized at a root of unity ζ. The ﬁrst approach comes from [ABG]. The main result of [ABG, Part I] relates3 Rep0(Uζ) to the derived category of equivariant coh”