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.
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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.
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Presented by: Henning Krause
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Ko, H. Cohomology in Singular Blocks for a Quantum Group at a Root of Unity. Algebr Represent Theor 22, 1109–1132 (2019). https://doi.org/10.1007/s10468-018-9814-4
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DOI: https://doi.org/10.1007/s10468-018-9814-4