Lattices like the Leech lattice

The Leech lattice has many strange properties, discovered by Conway, Parker, and Sloane. For example, it has covering radius √ 2, and the orbits of points at distance at least √ 2 from all lattice points correspond to the Niemeier lattices other than the Leech lattice. (See Conway and Sloane [6, Chaps. 22-28].) Most of the properties of the Leech lattice follow from the fact that it is the Dynkin diagram of the Lorentzian lattice II25,1, as in Conway [4]. In this paper we show that several other well-known lattices, in particular the Barnes-Wall lattice and the Coxeter-Todd lattice (see [6, Chap. 4]) are related to Dynkin diagrams of reflection groups of Lorentzian lattices; all these lattices have properties similar to (but more complicated than) those of the Leech lattice. Conway and Norton [5] showed that there was a strange correspondence between some automorphisms of the Leech lattice, some elements of the monster, some sublattices of the Leech lattice and some of the sporadic simple groups. Many of these things also correspond to some Lorentzian lattices behaving like II25,1 and to some infinite dimensional Kac-Moody algebras. Most of the notation and terminology is standard. For proofs of the facts about the Leech lattice that we use, see the original papers of Conway, Parker and Sloane in chapters 23, 26, and 27 of Conway and Sloane [6], or see Borcherds [1]. Lattices are always integral, and usually positive definite or Lorentzian, although they are occasionally singular. A root of a lattice means a vector r of positive norm such that reflection in the hyperplane of r is an automorphism of the lattice and such that r is primitive, i.e., r is not a nontrivial multiple of some other lattice vector; by a strong root we mean a root r such that (r, r) divides (r, v) for all v in the lattice. (For example, any vector of norm 1 is a strong root.) The symbols an, bn, . . . , e8 stand for the spherical Dynkin diagrams, and their corresponding affine Dynkin diagrams are denoted by An, Bn, . . . , E8. In some of the examples we give later E8 also stand for the E8 lattice. The symbols In,1 and IIn,1 stand for odd and even unimodular Lorentzian lattices of dimension n+ 1, which are unique up to isomorphism. The automorphism group Aut(R) of a Lorentzian lattice R means the group of automorphisms that fix each of the two cones of negative norm vectors (so Aut(R) has index 2 in the “full” automorphism group of R). The reflection group of a Lorentzian lattice is the group generated by the reflections of its roots. For any Lorentzian lattice, one of the two components of the norm −1 vectors can be identified with hyperbolic space, and all automorphisms of the lattice act as isometries on this space. In particular, a reflection of the lattice can be thought of as a reflection in hyperbolic space, so the reflection group of a lattice is a hyperbolic reflection group. Section 1 contains several results useful for practical calculation of Dynkin diagrams of Lorentzian lattices, Section 2 contains some results about the reflection group of a sublattice fixed by some group, and Section 3 applies the results of Sections 1 and 2 to the Leech lattice to produce several lattices whose reflection groups either have finite index