# Differential Equations and Quantum Groups

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#### pp: 255–268

#### DOI: 10.4171/020-1/11

^{[1]}(1) Department of Algebra and Geometry, Kolomna State Pedagogical Institute, Zelenaya, 30, 140411, KOLOMNA, RUSSIAN FEDERATION

A special class of flat logarithmic connections ∇_{R} on ℂ^{n} associated with finite complex vector configurations `R` ⊂ ℂ^{n} generating ℂ^{n} is considered. The connection ∇_{R} acts on the trivial holomorphic bundle with fiber ℂ^{n} and it has logarithmic poles on hyperplanes that
are orthogonal to vectors of `R` with respect to the standard Hermitian form of ℂ^{n}. We prove that Veselov’s ∨-conditions for a complex vector configuration `R` are equivalent to the Frobenius integrability of the connection ∇_{R}. If `R` is a root system with finite complex reflection group `W`(`R`), then `R` satisfies Veselov’s conditions and ∇_{R} is an integrable connection. In the case of some root systems `R`, we describe the monodromy representation of the generalized braid group `B`_{n}(`R`) defined by the associated logarithmic connection ∇_{R} on the quotient space ℂ^{n}/`W`(`R`). These representations are deformations of the standard representations of the corresponding complex reflection groups. They are generalizations of the Burau representations for some complex root systems which were earlier defined by Squier and Givental only for real root systems.

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