Explicit Riemann Hilbert correspondence

Question

For simplicity, we assume that $X=\mathbb P_{\mathbb C}^1-\{s_1, s_2, \dots, s_k\}$ and $\infty \in X$. Consider the trivial bundle $E=\mathcal O_X^r$ with the connection $\nabla$ induced by a Fushcian system on $X$, i.e. $$\nabla:= d+\sum_{i=1}^k \frac{A_i}{z-s_i},$$ where $A_i$'s are $r\times r$ constant matrices over $\mathbb C$ such that $\sum_{i=1}^kA_i=0$. Then we have a flat bundle with connection (or a logarithmic connection), note it only has regular singularities. My question is that what is the corresponding monodromy representation to $(E,\nabla)$ via the Riemann Hilbert correspondence? The known part is that locally at each $z=s_i$, the monodromy $T_i$ can be obtained by the residue map: $T_i=\exp(2\pi i A_i)$, but this seems not to admit a unique representation up to isomorphism. Is there any more properties/facts around the question?

Answer 1

It depends on what is called "explicit". If $k>2$, monodromy representation is a transcendental function of the $A_j$ and $s_j$. When $d=0$, it was expressed as an everywhere convergent power series in the $A_j$ whose coefficients are explicit (rational) functions in $s_j$ by Lappo-Danilevski:

Lappo-Danilevsky, J. A. Mémoires sur la théorie des systèmes des équations différentielles linéaires. (French) Chelsea Publishing Co., New York, N. Y., 1953.

If $d\neq 0$, the system is not Fuchsian: singularity at $\infty$ is irregular. If $k=2, d\neq 0$ your system already contains the "prolate/oblate spheroid equations", which were studied much and no reasonable explicit formula for the monodromy is known. There are asymptotics, of course. A recent paper about this special case, with a good reference list is

Richard-Jung, F.; Ramis, J.-P.; Thomann, J.; Fauvet, F. New characterizations for the eigenvalues of the prolate spheroidal wave equation. Stud. Appl. Math. 138 (2017), no. 1, 3–42.

The case $d=0$, $k=3$ was also much studied. The papers usually refer to "Heun's equation".

Answer 2

You are correct that the monodromy representation is given by the $T_i$. To address your concerns about this being unique up to isomorphism, notice that a change of basis of $\mathcal{O}_X^r$ induces a corresponding (compatible) change to the $T_i$.

You might be interested in the following references: Sections 5.1.1 and 5.1.2 of Hotta, Tanasaki, and Takeuchi's "D-Modules, Perverse Sheaves, and Representation Theory"; and Chapter III of "Algebraic D-Modules" by Borel, et al.