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

TL;DR

This paper establishes criteria for when line bundles on products of flag varieties descend to GIT quotients under the diagonal action of a simple algebraic group, linking descent to lattice conditions related to the group's root and weight lattices.

Contribution

It provides explicit lattice-based conditions involving the root and weight lattices that determine the descent of line bundles to GIT quotients for products of flag varieties.

Findings

Line bundles descend if weights are in multiples of the least common multiple of root coefficients and their sum lies in a specific sublattice.

Line bundles do not descend if the sum of weights is outside the root lattice.

The descent criteria depend only on the type of the Lie algebra of the group.

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 $\lambda + \mu + \nu \notin Q$.