Let $G$ be a connected, simplyconnected 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.
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1$\begingroup$ Might be it is discussed in Pressly, Segal "Loop groups". Let me just mention informal idea. G  is simple, so there is nontrivial element in H^3(G), by the trangression for the loops space it is mapped into H^2(LG)  the same cocycle which describes the central extension of LG. It is somehow invariant so we can push it down to G((t))/G[[t]]. So you get element in H^2(Gr). Now standard correspondence between H^2 and Line bundles gives you this line bundle. $\endgroup$ – Alexander Chervov Dec 30 '13 at 12:24

1$\begingroup$ The other way to describe having a cocycle of G((t)) you can construct a line bundle on G((t)) and then check its invariance. $\endgroup$ – Alexander Chervov Dec 30 '13 at 12:25

1$\begingroup$ See section 3.3 (and appendix A) of the recent PhD thesis of Brandon Levin (at his IAS webpage) for any reductive group $G$ (with connected fibers) over a Dedekind domain $A$; this fleshes out details sketched in notes of Gaitsgory and Faltings' paper "Algebraic loop groups...". Your case is $A=k[\![t]\!]$ for a field $k$, for which arguments simplify; when $G$ arises over $k\subset k[\![t]\!]$ then additional simplifications occur. A choice of $G\hookrightarrow {\rm{GL}}(V)$ is used to build the ample line bundle. (NB. You cannot "include" $\mathcal{G}r$ in a projective space; it is too big.) $\endgroup$ – user76758 Dec 30 '13 at 14:40

$\begingroup$ Thanks @Alexander Chervov and @user76758! Yes, I was definitely not referring to one of the projective spaces $\mathbb{P}^n$. I was referring to the projective space associated with some infinitedimensional complex vector space. In this case, the vector space is the dual of the space of global sections of the line bundle in question. $\endgroup$ – Peter Crooks Dec 30 '13 at 14:46
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You may look at Proposition 13.2.19 in S. Kumar's book "KacMoody Groups, their Flag Varieties and Representation Theory".