Let $G$ be a connected, simply-connected complex semisimple group. Let $$\mathcal{G}r:=G((t))/G[[t]]$$ be its affine Grassmannian. I have read that $\mathcal{G}r$ possesses a natural very ample line bundle/invertible sheaf (see "A Polytope Calculus for Semisimple Groups" by J. Anderson, for instance). This allows one to include $\mathcal{G}r$ in a projective space. I would very much appreciate a reference that explicitly describes this line bundle. 



Reference for the Natural Ample Line Bundle on the Affine Grassmannian
You may look at Proposition 13.2.19 in S. Kumar's book "Kac-Moody Groups, their Flag Varieties and Representation Theory".


