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We study a class of scalar, linear, non-local Riemann-Hilbert problems (RHP)involving finite subgroups of PSL(2,C). We associate to such problems a (maybeinfinite) root system and describe the relevance of the orbits of the Weylgroup in the construction of its solutions. As an application, we study indetail the large N expansion of SU(N) or SO(N) or Sp(2N) Chern-Simons partitionfunction Z_N(M) of 3-manifolds M that are either rational homology spheres ormore generally Seifert fibered spaces. It has a matrix model-likerepresentation, whose spectral curve can be characterized in terms of a RHP asabove. When pi_1(M) is finite (i.e. for manifolds M that are quotients of\mathbb{S}_{3} by a finite isometry group of type ADE), the Weyl groupassociated to the RHP is finite and the spectral curve is algebraic and can bein principle computed. We then show that the large $N$ expansion of Z_N(M) iscomputed by the topological recursion. This has consequences for theanalyticity properties of SU/SO/Sp perturbative invariants of knots alongfibers in $M$.

Root systems, spectral curves, and analysis of a Chern-Simons matrix model for Seifert fibered spaces