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.
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asked Dec 29, 2013 at 21:01
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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 non-trivial 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 ChervovCommented Dec 30, 2013 at 12:24
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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 ChervovCommented Dec 30, 2013 at 12:25
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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.) $\endgroup$– user76758Commented Dec 30, 2013 at 14:40
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$\begingroup$ 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 infinite-dimensional 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 CrooksCommented Dec 30, 2013 at 14:46
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You may look at Proposition 13.2.19 in S. Kumar's book "Kac-Moody Groups, their Flag Varieties and Representation Theory".
