Some Explicit Solutions of the Additive Deligne-Simpson Problem and Their Applications (Preprint)

Abstract

In this paper we construct three infinite series and two extra triples of complex matrices B, C, and A=B+C of special spectral types associated to C. Simpson's classification in his paper ``Products of Matrices'' and a classification of multiple flag varieties with finitely many orbits of the diagonal action of the general linear group by P. Magyar, J. Weyman, and A. Zelevinsky. This enables us to construct Fuchsian systems of differential equations which generalize the hypergeometric equation of Gauss-Riemann. In a sense, they are the closest relatives of the famous equation, because their triples of spectral flags have finitely many orbits of the diagonal action of the general linear group in the space of solutions. We also construct a scalar product such that A, B, and C are self-adjoint with respect to it. In the context of Fuchsian systems, this scalar product becomes a monodromy invariant complex symmetric bilinear form in the space of solutions. When the eigenvalues of A, B, and C are real, the matrices and the scalar product become real as well. We find inequalities on the eigenvalues of A, B, and C which make the scalar product positive-definite. As proved by A. Klyachko, the spectra of three hermitian (or real symmetric) matrices B, C, and A=B+C form a polyhedral convex cone in the space of triple spectra. He also gave a recursive algorithm to generate inequalities describing the cone. The inequalities we obtain describe non-recursively some faces of the Klyachko cone.