On the Riemann–Hilbert correspondence for generalized Knizhnik–Zamolodchikov equations for different root systems

Abstract

The paper is devoted to the restricted Riemann–Hilbert problem in a class of generalized Knizhnik–Zamolodchikov equations. The Knizhnik–Zamolodchikov equation has singularities on the set of reflection hyperplanes of the root system An−1. I. Cherednik proposed to consider also similar systems of equations, associated to the other root systems and gave examples of corresponding physical models. The Cherednik systems now are called the generalized Knizhnik–Zamolodchikov equations. In this class of equations the Riemann–Hilbert problem consists in investigating the correspondence between the equations and the representations of the fundamental group of the complement of the singular locus of the equation. For the case of the root system An this correspondence was investigated by V. Drinfeld and T. Kohno. In the paper an exposition of the results on the generalization of the Drinfeld–Kohno theory for the root system Bn obtained in collaboration with V. P. Lexin is given. Some principal elements of the proof of the main theorems are discussed. Besides, the equations of the Knizhnik–Zamolodchikov type for the other root systems and some equations close to them are touched.