Say $X= \mathbb{P^1}\times \cdots \times \mathbb{P}^1$ is a product of $n\geq3$ lines. Let the group $G=\text{SL}(2)$ act on $X$ diagonally, and let $\mathcal{L} = \mathcal{L}(a\_1,\ldots,a\_n)$ be the $G$-equivariant line bundle corresponding to positive weights $a\_1,\ldots,a\_n$. Suppose $\mathcal{L}$ has $G$-invariant global sections. Then the GIT quotient $Y=X \mathbin{/\mkern-6mu/}\_\mathcal{L} G$ is nonempty.


Then $\mathcal{L}$ descends to $Y$, making $Y$ a polarized variety (here I'm a little unsure, but I think it's OK and certainly some power of $\mathcal{L}$ descends.) My question is: is there a nice formula for the volume of $(Y,\mathcal{L})$, by which I mean the self-intersection number $\mathcal{L}^{dim(Y)}$? By nice I mean expressed in terms of the representation theory of $G$ and the weights $a\_i$.


The 'quantum' version of this question is the computation of the volume of the moduli space of parabolic $\text{SL}(2)$ bundles over a curve, and is given by Witten's volume formula. Taking the level high enough (and letting the curve be $\mathbb{P}^1$) I believe one gets a formula for the volume I'm looking for, but I was hoping for a more elementary formula in the 'classical' case. At the very least I would like a formula that doesn't involve the level at all.



Volume of a GIT quotient of projective lines
I don't know quite what would count as a nice formula. Here are a couple:


1. Consider the polytope of $(n-3)$-tuples $(d\_1,\ldots,d\_{n-3})$ of nonnegative numbers, such that $(a\_1,a\_2,d\_1),(d\_{n-3},a\_{n-1},a\_n)$ and each $(d\_i,a\_{i+2},d\_{i+1})$ are triples satisfying the triangle inequality. Then the volume of this polytope answers your question.
2. Let $s=\frac{\sum a\_i}{2}$. Then there is a formula for the multiplicity of the weight $\vec a$ in the irrep $V\_{s \omega\_2}$ of $GL\_n$, when $s$ is integral. The leading order behavior of this inhomogeneous piecewise polynomial in $\vec a$ also answers your question.


Actually the second, plus the Gel'fand-Cetlin system, gives the first, as I proved in my thesis. The first is the Verlinde formula in this genus zero setting.


