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


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.


Root of unity quantum group Cohomology Kazhdan-Lusztig polynomial 

Mathematics Subject Classification (2010)

17B37 20G05 



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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Authors and Affiliations

  1. 1.Max-Planck Institute for MathematicsBonnGermany

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