An investigation into Lie algebra representations obtained from regular holonomic D-modules

TL;DR

This paper explores the connection between Lie algebra representations and holonomic D-modules on the projective line, providing a topological perspective, extending previous examples, and developing computational tools for broader cases.

Contribution

It offers a topological description of Lie algebra representations from D-modules, generalizes to multiple singularities, and introduces a computer program for computations.

Findings

Characterization of sl_2-representations with up to 2 singularities

Construction of examples with more singularities

Development of a computational tool for the correspondence

Abstract

Beilinson--Bernstein localisation relates representations of a Lie algebra $\mathfrak{g}$ to certain $\mathcal{D}$-modules on the flag variety of $\mathfrak{g}$. In [arXiv:2002.01540], examples of $\mathfrak{sl}_2$-representations which correspond to $\mathcal{D}$-modules on $\mathbb{CP}^1$ were computed. In this expository article, we give a topological description of these and extended examples via the Riemann-Hilbert correspondence. We generalise this to a full characterisation of $\mathfrak{sl}_2$-representations which correspond to holonomic $\mathcal{D}$-modules on $\mathbb{CP}^1$ with at most 2 regular singularities. We construct further examples with more singularities and develop a computer program for the computation of this correspondence in more general cases.