On the Deligne-Simpson problem and its weak version

Abstract

We consider the {\em Deligne-Simpson problem (DSP) (resp. the weak DSP): Give necessary and sufficient conditions upon the choice of the $p+1$ conjugacy classes $c_j\subset gl(n,{\bf C})$ or $C_j\subset GL(n,{\bf C})$ so that there exist irreducible $(p+1)$-tuples (resp. $(p+1)$-tuples with trivial centralizers) of matrices $A_j\in c_j$ with zero sum or of matrices $M_j\in C_j$ whose product is $I$.} The matrices $A_j$ (resp. $M_j$) are interpreted as matrices-residua of Fuchsian linear systems (resp. as monodromy matrices of regular linear systems) of differential equations with complex time. In the paper we give sufficient conditions for solvability of the DSP in the case when one of the matrices is with distinct eigenvalues.