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

TL;DR

This paper extends the calculation of Ext groups in quantum group modules to singular blocks using parabolic Kazhdan-Lusztig polynomials, revealing new cohomological structures and parity properties.

Contribution

It provides explicit formulas for Ext dimensions in singular blocks of quantum groups, generalizing previous regular case results with new polynomial tools.

Findings

Ext dimensions are determined by parabolic Kazhdan-Lusztig polynomial coefficients.

Cohomology computations are extended to q-Schur and generalized q-Schur algebras.

A parity vanishing property is established via Kazhdan-Lusztig correspondence and Koszul grading.

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 $L,L'$ are irreducible $U_\zeta$-modules having regular highest weights, the dimension of $\operatorname{Ext}^n_{U_\zeta}(L,L')$ 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 $L,L'$ irreducible modules in a singular block that $\dim\operatorname{Ext}^n_{U_\zeta}(L,L')$ 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.