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Some conditions for descent of line bundles to GIT quotients $(G/B \times G/B \times G/B)//G$

We consider the descent of line bundles to GIT quotients of products of flag varieties. Let $G$ be a simple, connected, algebraic group over $\mathbb{C}$. We fix a Borel subgroup $B$ and consider the diagonal action of $G$ on the projective variety $X = G/B \times G/B \times G/B$. For any triple $(λ, μ, ν)$ of dominant regular characters there is a $G$-equivariant line bundle $\mathcal{L}$ on $X$. Then, $\mathcal{L}$ is said to descend to the GIT quotient $π:[X(\mathcal{L})]^{ss} \rightarrow X(\mathcal{L})//G$ if there exists a line bundle $\hat{\mathcal{L}}$ on $X(\mathcal{L})//G$ such that $\mathcal{L}\mid_{[X(\mathcal{L})]^{ss}} \cong π^*\hat{\mathcal{L}}$. 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 $\mathfrak{g}$ of $G$ written in terms of simple roots. We show that $\mathcal{L}$ descends if $λ, μ, ν\in dΛ$ and $λ+ μ+ ν\in Γ$, where $Γ$ is a fixed sublattice of $Q$ depending only on the type of $\mathfrak{g}$. Moreover, $\mathcal{L}$ never descends if $λ+ μ+ ν\notin Q$.