Let $X$ be a smooth proper algebraic curve over $\mathbb{C}$, and let $G$ be a reductive group over $\mathbb{C}$. Let $Gr_{X,n}$ be the Beilinson-Drinfeld Grassmannian (for n points in $X$), which classifies triples $\left( (x_i) \in X^n, \mathcal{P}, \beta \right)$, where $(x_i)$ is an $n$-tuple of points in $X$, $\mathcal{P}$ is a $G$-bundle on $X$, and $\beta$ is a trivialization of $\mathcal{P}$ away from the set $\{x_i\}$. We have a natural map $Gr_{X,n} \rightarrow \text{Bun}_G(X)$ given by remembering only $\mathcal{P}$. Here $\text{Bun}_G(X)$ is the moduli stack of $G$-bundles on $X$. This stack carries a natural line bundle $\mathcal{L}$ called the determinant line bundle. My question is what are the global sections of $\mathcal{L}$ restricted to $Gr_{X,n}$?
If instead of the Beilinson-Drinfeld Grassmannian, I look at the ordinary affine Grassmannian (which classifies $G$-bundles on a curve trivialized away from a fixed point), the answer to this question is the following: global sections of $\mathcal{L}$ are given by $V(\Lambda_0)^*$, the dual of the basic representation of the affine Kac-Moody group associated to $G$. I want to know whether there is an answer to my question in terms of representation theory of the affine Lie algebra.
Let me say why I ask this question. I am interested in the following situation: we can consider the subspace of $Gr_{X,2}$ where the second point $x_2$ is fixed. This space maps to $X$ by remembering the first point, and this map is known to be flat. If we look at the fiber of this map over any point except $x_2$, we get a product of two copies of the affine Grassmannian. If the first point coincides with $x_2$, we get a single copy of the affine Grassmannian. Because of flatness of the map to $X$ and a higher cohomology vanishing statement, on the level of sections of the determinant bundle, we expect to find that $V(\Lambda_0)^* \otimes V(\Lambda_0)^* \cong V(\Lambda_0)^*$ (c.f. Theorem 1.2.2 in the following paper (http://arxiv.org/abs/0710.5247) by Xinwen Zhu where this construction is carried out geometrically for Demazure modules, where everything is finite dimensional). Is there any technical problem in using the techniques of this paper to conclude that $V(\Lambda_0)^* \otimes V(\Lambda_0)^* \cong V(\Lambda_0)^*$ (perhaps with some completion), and if so, can this map be algebraically understood? I was hoping that understanding global sections on the full Beilinson-Drinfeld Grassmannian would be a step towards understanding this.
