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Algebras and Representation Theory

, Volume 22, Issue 5, pp 1109–1132 | Cite as

Cohomology in Singular Blocks for a Quantum Group at a Root of Unity

  • Hankyung KoEmail author
Open Access
Article

Abstract

Let Uζ be a Lusztig quantum enveloping algebra associated to a complex semisimple Lie algebra \(\mathfrak {g}\) and a root of unity ζ. When L, L are irreducible Uζ-modules having regular highest weights, the dimension of \(\text {Ext}^{n}_{U_{\zeta }}(L,L^{\prime })\) can be calculated in terms of the coefficients of appropriate Kazhdan-Lusztig polynomials associated to the affine Weyl group of Uζ. This paper shows for L, L irreducible modules in a singular block that \(\dim \text {Ext}^{n}_{U_{\zeta }}(L,L^{\prime })\) is explicitly determined using the coefficients of parabolic Kazhdan-Lusztig polynomials. This also computes the corresponding cohomology for q-Schur algebras and many generalized q-Schur algebras. The result depends on a certain parity vanishing property which we obtain from the Kazhdan-Lusztig correspondence and a Koszul grading of Shan-Varagnolo-Vasserot for the corresponding affine Lie algebra.

Keywords

Root of unity quantum group Cohomology Kazhdan-Lusztig polynomial 

Mathematics Subject Classification (2010)

17B37 20G05 

Notes

Acknowledgements

The author thanks Brian Parshall and Leonard Scott for explaining their conjecture and related subjects to her, pointing out errors in previous proofs, encouraging her to write this into a paper, and carefully reading several versions of this paper.

Funding Information

Open access funding provided by Max Planck Society.

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Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Max-Planck Institute for MathematicsBonnGermany

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