0 Some examples related to the Deligne - Simpson problem ∗

In the present paper we consider some examples relative to the Deligne-Simpson problem (DSP) which is formulated like this: Give necessary and sufficient conditions upon the choice of the p+ 1 conjugacy classes cj ⊂ gl(n,C), resp. Cj ⊂ GL(n,C), so that there exist irreducible (p+1)-tuples of matrices Aj ∈ cj , A1 + . . .+Ap+1 = 0, resp. of matrices Mj ∈ Cj , M1 . . .Mp+1 = I. By definition, the weak DSP is the DSP in which the requirement of irreducibility is replaced by the weaker requirement the centralizer of the (p + 1)-tuple of matrices to be trivial. The matrices Aj, resp. Mj , are interpreted as matrices-residua of Fuchsian systems on Riemann’s sphere (i.e. linear systems of ordinary differential equations with logarithmic poles), resp. as monodromy operators of regular systems on Riemann’s sphere (i.e. linear systems of ordinary differential equations with moderate growth rate of the solutions at the poles). Fuchsian systems are a particular case of regular ones. By definition, the monodromy operators generate the monodromy group of a regular system. In the multiplicative version (i.e. for matrices Mj) the classes Cj are interpreted as local monodromies around the poles and the problem admits the interpretation: For what (p + 1)-tuples of local monodromies do there exist monodromy groups with such local monodromies.