Mathematics

A complex crystallographic (CC) group Γ is a discrete group of affine transformations of the complex space C^n acting with a compact quotient. Any such group is an extension of a finite linear group G, called the point group, by a lattice L of maximal rank 2n. A CC group is of reflection type (a CCR group) if it is generated by affine reflections. A conjecture of Bernstein-Schwarzman suggests that the quotient C^n/Γ is a weighted projective space when Γ is irreducible; this is a natural generalization of the Shephard-Todd-Chevalley theorem for finite linear groups generated by reflections. The conjecture is known in dimension 2 and for CCR groups of Coxeter type, that is those whose point group G is conjugate to a real Coxeter group. In the talk, the case of a genuinely complex CCR group Γ in dimension 3 will be discussed, with quasi-simple point group G of order 336. In this case, C^n/Γ can be interpreted as the quotient of the Jacobian of Klein's quartic curve by its full automorphism group {±1}×H, where H is Klein's simple group of order 168.