Discrete groups generated by complex reflections. Lecture 1

An affine isometry of a complex Hermitian space $E$ is called a complex reflection if its order is finite and the codimension of the fixed point set (the mirror of reflection) is equal to $1$. These lectures will be devoted to the transformation groups $G$ of the space $E$, which are generated by reflections and discrete (the latter means that the $G$-orbit of every point $x$ of $E$ is a discrete subset of $E$ and the $G$-stabilizer of $x$ is finite). For instance, if $E={\mathbf C}^1$, then the cyclic group of order $n$ consisting of all rotations about zero through the $2\pi/n$-multiple angles, is a finite such group; it is generated by a single reflection. In this example, $E/G$ is a noncompact algebraic variety (isomorphic to the affine line ${\mathbf C}^1$). There are also infinite discrete groups generated by complex reflections: for instance, such is the group generated by rotations through the $2\pi/3$-multiple angles about the points of the lattice $\mathbf Z+e^{2\pi i/3}\mathbf Z$ of equilateral triangles in $E={\mathbf C}^1$. For it, the quotient $E/G$ is a compact algebraic variety (isomorphic to the projective line ${\mathbf P}^1$). In these lectures, we will describe the classification of discrete groups generated by complex reflections and dwell on the various remarkable objects appearing in the context of this theory, in particular, on the invariant lattices and complex spaces $E/G$ (which always turn out to be algebraic varieties).

Lecture 1.
Formulation of the problem. The real and complex reflection groups. Examples: one-dimensional complex reflection groups. Two classes: finite and infinite reflection groups. Reduction to the irreducible case. Classification of the real reflection groups.