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

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.

arXiv commentExpository article in preliminary form; v2: corrections made in section 5