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

Abstract

Let $U_{\zeta }$ be a Lusztig quantum enveloping algebra associated to a complex semisimple Lie algebra $\mathfrak{g}$ and a root of unity $\zeta$. When $L,L^{\prime }$ are irreducible $U_{\zeta }$-modules having regular highest weights, the dimension of ${\mathrm{Ext}}_{U_{\zeta }}^n(L,L^{\prime })$ can be calculated in terms of the coefficients of appropriate Kazhdan-Lusztig polynomials associated to the affine Weyl group of $U_{\zeta }$. This paper shows for $L,L^{\prime }$ irreducible modules in a singular block that $\dim {\mathrm{Ext}}_{U_{\zeta }}^n(L,L^{\prime })$ 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.

minor corrections made. v3 is the accepted version

References in corpus (2)

• Q-Koszul Algebras and Three Conjectures
• Remarks on the ABG Induction Theorem

Cited by in corpus (2)

• On modules arising from quantum groups at $p^r$-th roots of unity
• Grade zero part of forced graded algebras