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

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 $X=G\mathrm{/}B\times G\mathrm{/}B\times G\mathrm{/}B$. For any triple $(\lambda, \mu, \nu)$ofdominantregularcharactersthereisa$G$-equivariant line bundle $\mathcal{L}$ on $X$. Then, $\mathcal{L}$ is said to descend to the GIT quotient $\pi :[X(\mathcal{L}){]}^{ss}\to X(\mathcal{L})\mathrm{/}\mathrm{/}G$ if there exists a line bundle $\widehat{\mathcal{L}}$ on $X(\mathcal{L})\mathrm{/}\mathrm{/}G$ such that $\mathcal{L}{\mid }_{[X(\mathcal{L}){]}^{ss}}\cong {\pi }^{\ast }\widehat{\mathcal{L}}$. Let $Q$ be the root lattice, $\mathrm{\Lambda }$ 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 $\lambda ,\mu ,\nu \in d\mathrm{\Lambda }$ and $\lambda + \mu + \nu \in \Gamma$,where$\Gamma$isafixedsublatticeof$Q$ depending only on the type of $\mathfrak{g}$. Moreover, $\mathcal{L}$ never descends if $\lambda

• \mu + \nu \notin Q$.