On the Deligne-Simpson problem and its weak version

2003/10/28 by Vladimir Petrov Kostov
Mathematics · #math.AG #math.RT #msc:15A24

paper · pdf

published as Bulletin des Sciences Mathematiques 128/2 (2004) 105-125 · 22 pages

arxiv created 2003/10/28 · arxiv updated 2009/12/01

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⊂ gl(n,\bf C) or C_j⊂ GL(n,\bf C) so that there exist irreducible (p+1)-tuples (resp. (p+1)-tuples with trivial centralizers) of matrices A_j∈ c_j with zero sum or of matrices M_j∈ 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.

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