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

Abstract: 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 $(\lambda, \mu, \nu)$ 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 $\pi:[X(\mathcal{L})]^{ss} \rightarrow X(\mathcal{L})//G$ if there exists a line bundle $\hat{\mathcal{L}}$ on $X(\mathcal{L})//G$ such that $\mathcal{L}\mid_{[X(\mathcal{L})]^{ss}} \cong \pi^*\hat{\mathcal{L}}$. Let $Q$ be the root lattice, $\Lambda$ the weight lattice, and $d$ the least common multiple of the coefficients of the highest root $\theta$ of the Lie algebra $\mathfrak{g}$ of $G$ written in terms of simple roots. We show that $\mathcal{L}$ descends if $\lambda, \mu, \nu \in d\Lambda$ and $\lambda + \mu + \nu \in \Gamma$, where $\Gamma$ is a fixed sublattice of $Q$ depending only on the type of $\mathfrak{g}$. Moreover, $\mathcal{L}$ never descends if $\lambda + \mu + \nu \notin Q$.