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

We consider the descent of line bundles to GIT quotients of products of flag varieties. Let G be a simple, connected, algebraic group over C. We fix a Borel subgroup B and consider the diagonal action of G on the projective variety X=G/B×G/B×G/B. For any triple (λ,μ,ν) of dominant regular characters there is a G-equivariant line bundle L on X. Then, L is said to descend to the GIT quotient π:X^ss(L)→X(L)//G if there exists a line bundle Lˆ on X(L)//G such that L|_X^ss(L)≅π^⁎Lˆ. 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.