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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.

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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.

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  • $\begingroup$ Thanks for your answer! About #2: What is the intuition behind looking at the multiplicity in $\vec{a}$ in $V_{s\omega_2}$? (I'm assuming $\omega_2$ is the second fundamental weight.) Is the formula the Verlinde formula or is it something else? A reference would be much appreciated! $\endgroup$
    – Michael Schuster
    Commented Feb 10, 2015 at 15:44
  • $\begingroup$ Take $M_{2\times n}$ with $GL(2)$ acting on the left, $T^n \leq GL(n)$ acting on the right. If you GIT quotient right then left, you get your space (or really, Deligne-Mostow's). If you GIT quotient left then right, you get the weight space interpretation I'm talking about. Check out these papers (in reverse order): front.math.ucdavis.edu/search?a=knutson&t=polygon $\endgroup$
    – Allen Knutson
    Commented Feb 10, 2015 at 19:53

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