Last update: 12 May 2014
This is an excerpt of the lecture notes Discrete complex reflection groups by V.L. Popov. Lectures delivered at the Mathematical Institute, Rijksuniversiteit Utrecht, October 1980.
These notes essentially record the content of a course of five lectures given at the Mathematical Institute of the Rijksuniversiteit Utrecht in October 1980. The aim of these lectures was to develop the theory of discrete groups generated by affine unitary reflections, and in fact, to provide a classification of these groups. I made an attempt to give an exposition in such a way that our results would be comparable with the classical results of the theory of discrete groups generated by affine reflections in a real eUclidian space, which was developed in the former half of this century by Coxeter, Witt, Stiefel and others. Both theories have much in common. However, the classification of complex groups is more complicated (and less geometrical) than the classification over the reals. The problem of developing the theory of discrete groups generated by affine unitary reflections is a comparatively old one; I was informed that it was posed by A. Borel about 15 years ago. My general aim during the lectures was to explain the main ideas and to give proofs only of the theorems that are not of a strictly technical nature therefore, I restricted myself to examples or simply to formulating results in all technical cases (which are, however, not always trivial). A more detailed exposition will appear elsewhere.
I wish to thank the Mathematical Institute of the Rijksuniversiteit Utrecht for its hospitality. I am grateful to Professor T.A. Springer on whose initiative these lectures were given and written up. I am also grateful to A.M. Cohen for the interesting discussions I had with him and for his help on preparing these notes. My thanks go also to the secretaries of the University of Utrecht, the Netherlands for the careful typing of the manuscript.