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

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.