Ising vectors in the vertex operator algebra V⁺∧ associated with the leech lattice ∧

In this article, we study the Ising vectors in the vertex operator algebra V⁺∧ associated with the Leech lattice ∧. The main result is a characterization of the Ising vectors in V⁺∧. We show that for any Ising vector e in V⁺∧, there is a sublattice E ≅ √2E₈ of ∧ such that e ∈ V⁺E . Some properties about their corresponding τ -involutions in the moonshine vertex operator algebra V are also discussed.We show that there is no Ising vector of σ- type in V. Moreover, we compute the centralizer CAutV(z, τe) for any Ising vector e ∈ V⁺∧, where z is a 2B element in AutV which fixes V⁺∧. On the basis of this result, we also obtain an explanation for the 1A case of an observation by Glauberman-Norton (2001), which describes some mysterious relations between the centralizer of z and some 2A elements commuting z in the Monster and the Weyl groups of certain sublattices of the root lattice of type E₈.