On the Deligne--Simpson Problem

V. P. Kostov

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Monodromy in problems of algebraic geometry and differential equations, Tome 238 (2002), pp. 158-195

Résumé

The Deligne–Simpson problem is formulated as follows: \textit{give necessary and sufficient conditions for the choice of the conjugacy classes $C_j\subset SL(n,\mathbb C)$ or $c_j\subset sl(n,\mathbb C)$ so that there exist irreducible $(p+1)$-tuples of matrices $M_j\in C_j$ or $A_j\in c_j$ satisfying the equality $M_1\ldots M_{p+1}=I$ or $A_1+\ldots +A_{p+1}=0$}. We solve the problem for generic eigenvalues with the exception of the case of matrices $M_j$ when the greatest common divisor of the numbers $\Sigma _{j,l}(\sigma )$ of Jordan blocks of a given matrix $M_j$, with a given eigenvalue $\sigma$ and of a given size $l$ (taken over all $j$, $\sigma$, $l$), is $>1$. Generic eigenvalues are defined by explicit algebraic inequalities. For such eigenvalues, there exist no reducible $(p+1)$-tuples. The matrices $M_j$ and $A_j$ are interpreted as monodromy operators of regular linear systems and as matrices–residua of Fuchsian ones on Riemann's sphere.