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Journal Article

Cohomology in singular blocks for a quantum group at a root of unity


Ko,  Hankyung
Max Planck Institute for Mathematics, Max Planck Society;

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Ko, H. (2019). Cohomology in singular blocks for a quantum group at a root of unity. Algebras and Representation Theory, 22(5), 1109-1132. doi:10.1007/s10468-018-9814-4.

Cite as: https://hdl.handle.net/21.11116/0000-0005-1EE5-8
Let Uζ be a Lusztig quantum enveloping algebra associated to a complex semisimple Lie algebra g and a root of unity ζ. When L, L′ are irreducible Uζ-modules having regular highest weights, the dimension of ExtnUζ(L,L′) 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 dimExtnUζ(L,L′) 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.