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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
CHEVIE
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