arXiv subjects

Hankyung Ko

Publications and source records attributed to Hankyung Ko.

21 records · Page 2Linked to original sources

arXiv2018

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

Let $U_ζ$ 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 $U_ζ$-modules having regular highest weights, the dimension of $\operatorname{Ext}^n_{U_ζ}(L,L')$ can be calculated in terms of the coefficients of appropriate Kazhdan-Lusztig polynomials associated to the affine Weyl group of $U_ζ$. This paper shows for $L,L'$ irreducible modules in a singular block that $\dim\operatorname{Ext}^n_{U_ζ}(L,L')$ 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.

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arXiv2017

Grade zero part of forced graded algebras

The paper concerns a certain subcategory of the category of representations for a semisimple algebraic group $G$ in characteristic $p$, which arise from the semisimple modules for the corresponding quantum group at a $p$-th root of unity. The subcategory, thus, records the cohomological difference between quantum groups and algebraic groups. We define translation functors in this category and use them to obtain information on the irreducible characters for $G$ when the Lusztig character formula does not hold.

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arXiv2016

On modules arising from quantum groups at $p^r$-th roots of unity

This paper studies the "reduction mod $p$" method, which constructs large classes of representations for a semisimple algebraic group $G$ from representations for the corresponding Lusztig quantum group $U_ζ$ at a $p^r$-th root of unity. The $G$-modules arising in this way include the Weyl modules, the induced modules, and various reduced versions of these modules. We present a relation between $\operatorname{Ext}^n_G(V,W)$ and $\operatorname{Ext}^n_{U_ζ}(V',W')$, when $V,W$ are obtained from $V',W'$ by reduction mod $p$. Since the dimensions of $\operatorname{Ext}^n$-spaces for $U_ζ$-modules are known in many cases, our result guarantees the existence of many new extension classes and homomorphisms between certain rational $G$-modules. One application is a new proof of James Franklin's result on certain homomorphisms between two Weyl modules. We also provide some examples which show that the $p$-th root of unity case and a general $p^r$-th root of unity case are essentially different.

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