The Deligne-Simpson problem for zero index of rigidity

Abstract

We consider the {\em Deligne-Simpson problem}: {\em Give necessary and sufficient conditions for the choice of the conjugacy classes $c_j\subset gl(n,{\bf C})$ or $C_j\subset GL(n,{\bf C})$, $j=1,..., p+1$, so that there exist irreducible $(p+1)$-tuples of matrices $A_j\in c_j$ whose sum is 0 or of matrices $M_j\in C_j$ whose product is $I$.} The matrices $A_j$ (resp. $M_j$) are interepreted as matrices-residua of Fuchsian linear systems (resp. as monodromy operators of regular systems) on Riemann's sphere. We consider the case when the sum of the dimensions of the conjugacy classes $c_j$ or $C_j$ is $2n^2$ and we prove a theorem of non-existence of such irreducible $(p+1)$-tuples.