Involutions in the affine Conway group

Abstract

The affine Conway group is the group of affine isometries of the Leech lattice. This group is isomorphic to the automorphism group of a standard fundamental domain for the action of the Weyl group on the even hyperbolic lattice II_{1, 25} of rank 26. In this paper, we show that the affine Conway group has exactly nine conjugacy classes of involutions. We investigate their properties and show that, among these nine classes, one class can be regarded as an analogue of the class of Enriques involutions of K3 surfaces. Motivated by possible applications to K3 and Enriques surfaces, we investigate in detail the orthogonal groups of the hyperbolic lattices arising as the invariant sublattices of involutions in II_{1, 25}. This computation is carried out using the Borcherds method. Unlike the examples considered previously, the induced chambers possess infinitely many walls.