Is every complex linear algebraic group a differential Galois group?

Question

Let $ G $ be a complex linear algebraic group. In other words, $ G $ is a subvariety of the space of $ n \times n $ complex matrices and $ G $ is a group under matrix multiplication.

Does there always exist a differential equation for which $ G $ is the differential Galois group?

For example, $ \mathrm{SL}(2,\mathbb{C}) $ is the differential Galois group of $ E $ over $ K $ where $ K= \mathbb{C}(t) $ is the field of rational functions and $ E $ is the splitting field of the Airy differential operator $ u''(t)-tu(t) $, the source is theorem 4 of https://pi.math.cornell.edu/~hubbard/diffalg1.pdf

Theorem 3 of the same reference say that the differential Galois group of a linear differential equation is always a linear algebraic group. So I am really asking about the converse of Theorem 3: is every complex linear algebraic group the differential Galois group of some linear differential equation?

Answer 1

This was shown in 1979 by Carol Tretkoff and Marvin Tretkoff in Solution of the inverse problem of differential Galois theory in the classical case by analytic methods. In 2005 this was reproved by Julia Hartmann in On the inverse problem in differential Galois theory using algebraic methods, which allowed to replace the complex numbers by any algebraically closed field of characteristic $0$. Her work is based on previous work from 1996 by Mitschi and Singer who had the additional assumption that $G$ is connected.