Some conditions for descent of line bundles to GIT quotients (G/B × G/B × G/B)//G

Let Q be the root lattice, Λ the weight lattice, and d the least common multiple of the coefficients of the highest root θ of the Lie algebra g of G written in terms of simple roots. We show that L descends if λ,μ,ν∈dΛ and λ+μ+ν∈Γ, where Γ is a fixed sublattice of Q depending only on the type of g. Moreover, L never descends if λ+μ+ν∉Q.