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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.83.

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Quantum groups, the loop Grassmannian, and the Springer resolution
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by Sergey Arkhipov, Roman Bezrukavnikov and Victor Ginzburg
J. Amer. Math. Soc. 17 (2004), 595-678
Published electronically: April 13, 2004


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.
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Bibliographic Information
  • Sergey Arkhipov
  • Affiliation: Department of Mathematics, Yale University, 10 Hillhouse Avenue, New Haven, Connecticut 06520
  • Email:
  • Roman Bezrukavnikov
  • Affiliation: Department of Mathematics, Northwestern University, Evanston, Illinois 60208
  • MR Author ID: 347192
  • Email:
  • Victor Ginzburg
  • Affiliation: Department of Mathematics, University of Chicago, Chicago, Illinois 60637
  • Email:
  • Received by editor(s): April 20, 2003
  • Published electronically: April 13, 2004
  • © Copyright 2004 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 17 (2004), 595-678
  • MSC (2000): Primary 16S38; Secondary 14A22
  • DOI:
  • MathSciNet review: 2053952