The classification of orthogonally rigid $G_2$-local systems and related differential operators

Michael Dettweiler, Stefan Reiter
2014 Transactions of the American Mathematical Society  
We prove a criterion for a general self-adjoint differential operator of rank 7 to have its monodromy group inside the exceptional algebraic group G 2 (C). We then classify orthogonally rigid local systems of rank 7 on the punctured projective line whose monodromy is dense in the exceptional algebraic group G 2 (C). can be seen as rigidity relative to the larger group GL n ) but still strong enough to impose a lot of structure on L. By the work of N. Katz on the middle convolution functor MC χ
more » ... all rigid irreducible local systems L on the punctured line can be constructed by applying iteratively MC χ and tensor products with rank-1-sheaves to a rank-1-sheaf. For orthogonally rigid local systems with G 2 -monodromy we prove that there is a similar method of construction (cf. Theorem 6.1 below): Theorem 1.1. Let L be an orthogonally rigid C-local system on a punctured projective line P 1 \ {x 1 , . . . , x r } of rank 7 whose monodromy group is dense in the exceptional simple group G 2 . If L has nontrivial local monodromy at x 1 , . . . , x r , then r = 3, 4 and L can be constructed by applying iteratively a sequence of the following operations to a rank-1-system: • Middle convolutions MC χ , with varying χ. • Tensor products with rank-1-local systems. • Tensor operations like symmetric or alternating products. • 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.
doi:10.1090/s0002-9947-2014-06042-x fatcat:nrebeph3kvctniwhgxwpqgp6ri