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The classification of orthogonally rigid \(G_{2}\)-local systems and related differential operators. (English) Zbl 1312.32015
Let \(O_n\) denote the orthogonal group on an \(n\)-dimensional (complex) vector space. An irreducible orthogonally self-dual complex rank \(n\) local system \(\mathcal{L}\) on the punctured complex projective line \(\mathbb{P}^1 \smallsetminus \{ x_1, \dots, x_r \}\) is said to be orthogonally rigid if \[ \sum_{i = 1}^{r+1} \mathrm{codim} (C_{O_n}(g_i)) = 2 \dim (O_n), \] where \(C_{O_n}(g_i)\) is the centralizer of the local monodromy generator \(g_i\). The main result of this article is a complete classification of the orthogonally rigid local systems of rank \(7\) whose monodromy group is Zariski dense in \(G_2(\mathbb{C}) \subset O_7\). Specifically, the authors show:
Let \(\mathcal{L}\) be a rigid \(\mathbb{C}\)-local system on a punctured projective line \(\mathbb{P}^1 \smallsetminus \{ x_1, \dots, x_r \}\) of rank \(7\) whose monodromy group is dense in the exceptional simple group \(G_2\). If \(\mathcal{L}\) has nontrivial local monodromy at \(x_1, \dots , x_r\), then \(r=3,4\) and \(\mathcal{L}\) can be constructed by applying iteratively a sequence of the following operations to a rank-\(1\)-system: 0,5 cm
\(\bullet\)
middle convolutions \(\mathrm{M}_{\chi}\), with varying \(\chi\);
\(\bullet\)
tensor products with rank-\(1\)-local systems;
\(\bullet\)
tensor operations like symmetric or alternating products;
\(\bullet\)
pullbacks along rational functions.
Especially, each such local system which has quasi-unipotent monodromy is motivic, i.e., it arises from the variation of periods of a family of varieties over the punctured projective line.
The proof of the theorem is a mixture of case by case analysis and general arguments.

MSC:
32S40 Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects)
20G41 Exceptional groups
Software:
CHEVIE
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