Realizing Rings of Regular Functions via the Cohomology of Quantum Groups

TL;DR

This paper explores the cohomology of small quantum groups to realize rings of regular functions on flag varieties and orbit closures, extending classical results to the quantum and parabolic contexts.

Contribution

It generalizes the cohomology ring computations of small quantum groups to the parabolic and quantum setting, linking module global sections with quantum group cohomology.

Findings

Generalized cohomology ring calculations for small quantum groups.

Connected module multiplicities with extension group dimensions.

Provided formulas for realizing coordinate rings of orbit closures.

Abstract

Let $G$ be a complex reductive group and $P$ be a parabolic subgroup of $G$. In this paper the authors address questions involving the realization of the $G$-module of the global sections of the (twisted) cotangent bundle over the flag variety $G/P$ via the cohomology of the small quantum group. Our main results generalize the important computation of the cohomology ring for the small quantum group by Ginzburg and Kumar, and provides a generalization of well-known calculations by Kumar, Lauritzen, and Thomsen to the quantum case and the parabolic setting. As an application we answer the question (first posed by Friedlander and Parshall for Frobenius kernels) about the realization of coordinate rings of Richardson orbit closures for complex semisimple groups via quantum group cohomology. Formulas will be provided which relate the multiplicities of simple $G$-modules in the global sections with the dimensions of extension groups over the large quantum group.