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Rigid local systems and the multiplicative eigenvalue problem. (English) Zbl 1495.14015

A rank \(\ell\) local system of complex vector spaces on \(\mathbb{P}^1_{\mathbb{C}}\setminus S\) where \(S\) is a collection of \(s\) distinct points, is equivalent to an \(s\)-tuple of matrices in GL\((\ell,\mathbb{C})\) whose product is the identity matrix. Rank \(\ell\) local systems can then be identified with \(\ell\)-dimensional representations. Such a local system is irreducible if the corresponding representation is irreducible. It is called rigid if any other local system with the same conjugacy classes of local monodromies is isomorphic to it.
The author proposes a construction producing irreducible complex rigid local systems on \(\mathbb{P}^1_{\mathbb{C}}\setminus S\) via quantum Schubert calculus and strange duality. These local systems are unitary and arise from a study of vertices in the polytopes controlling the multiplicative eigenvalue problem for the special unitary groups SU\((n)\) (i.e., determination of the possible eigenvalues of a product of unitary matrices given the eigenvalues of the matrices). Strange duals of the simplest vertices of these polytopes are shown to give all possible unitary irreducible rigid local systems. As a consequence one obtains that the ranks of unitary irreducible rigid local systems, including those with finite global monodromy, on \(\mathbb{P}^1_{\mathbb{C}}\setminus S\) are bounded above if one fixes the cardinality \(s\) of the set \(S\) and requires the local monodromies to have orders dividing \(n\) for a fixed \(n\). The author answers a question of N. Katz by showing that there are no irreducible rigid local systems of rank greater than one, with finite global monodromy, all of whose local monodromies have orders dividing \(n\), when \(n\) is a prime number.
The author proves also that all unitary irreducible rigid local systems on \(\mathbb{P}^1_{\mathbb{C}}\setminus S\) with finite local monodromies arise as solutions to the Knizhnik-Zamolodchikov equations on conformal blocks for the special linear group and gives an inductive mechanism for determining all vertices in the multiplicative eigenvalue problem for SU\((n)\).

MSC:

14D05 Structure of families (Picard-Lefschetz, monodromy, etc.)
14D20 Algebraic moduli problems, moduli of vector bundles
32G34 Moduli and deformations for ordinary differential equations (e.g., Knizhnik-Zamolodchikov equation)
14H60 Vector bundles on curves and their moduli
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