Cohomology of twisted $\mathcal{D}$-modules on $\mathbb{P}^1$ obtained as extensions from $\mathbb{C}^{\times}$

TL;DR

This paper constructs and analyzes twisted $ abla$-modules on $ abla$-projective line, revealing their structure and cohomology as $ abla$-modules, advancing the understanding of $ abla$-module categories related to Lie algebra representations.

Contribution

It introduces a new class of twisted $ abla$-modules on $ abla$-projective line as extensions from local systems on $ abla^{ imes}$, with explicit structure and cohomology descriptions.

Findings

Sheaf cohomology groups are weight modules for $ abla_{ abla}$

Explicit presentations for non-highest/lowest weight modules

Illustrates $ abla$-module concepts and Beilinson-Bernstein equivalence

Abstract

We construct twisted $\mathcal{D}$-modules on the projective line $\mathbb{P}^1$ that are equivariant for the action of the diagonal torus subgroup of $SL_2$. In the most interesting case these arise as extensions from local systems on $\mathbb{C}^{\times}$. We discuss their subquotient structure. Their sheaf cohomology groups are weight modules for the Lie algebra $\mathfrak{sl}_2$. We also discuss their subquotient structure and in case these modules are not the familiar highest or lowest weight modules, we give an explicit presentation for them. Our computations illustrate some basic $\mathcal{D}$-module concepts and the Beilinson-Bernstein equivalence. They are the first step in a program that aims to describe categories of modules over semisimple and affine Kac-Moody Lie algebras that are next to highest (or lowest) weight via $\mathcal{D}$-modules on the flag variety.