TITLE: Several questions on semi infinite flag manifold
QUESTION [6 upvotes]: Let $G$ be a connected simply-connected Lie group correspondence to simple Lie algebra $g$,consider the loop algebra $g((t))$ and so called "Natural Borel subalgebra" $n[t,t^{-1}]\oplus h[t]$,denoted by $\mathfrak{b}$ and consider ind-group $G((t))$ associated to $g((t))$ and ind group $N_{-}((t))$ and $H[[t]]$ correspondence to completion of $n_{-}[t,t^{-1}]$ and $h[t]$ in $g((t))$.
Look at the "Natural Borel subgroup" $B$ correspondence to "natural borel subalgebra" $\mathfrak{b}$
It is equal to $N_{-}((t))$$H[[t]]$. Now consider the quotient $X:= G((t))/B$. Frenkel and Ben zvi claimed that $X$ can not be given a scheme or an ind-scheme structures. 
My first question is How to see this? For example, if $G=SL_2(\mathbb{C})$, how to see $X$ can not be given a scheme structure? Can one use analogue of Birkhoff decomposition to give a scheme structure(the $w$-translate of big "semi-inifnite" cell form an open affine cover and algebra of regular functions on these big cell should identify with contragradient Wakimoto modules)
Then they claimed that $X$ should be viewed as formal loop space of finite dimensional flag variety $G/B_{-}$, which is $Hom(Spec\mathbb{C}((t)),G/B_{-})$. I wonder whether this statement is equivalent to say they are "isomorphic" to each other(in what sense?) and how to prove this statement? 
Any hint and related comments is welcome. 
Thanks!

REPLY [6 votes]: About defining the (ind)scheme structure: working with particular strata is basically never a good way to do this. What you need in order to define an algebro-geometric object is to
define a functor from $Schemes$ to $Sets$ that it represents (it is enough to do it for affine schemes, i.e. it is enough to say what is an $R$-point of your space when $R$ is a ring). This is easy to do for semi-infinite flags. After you have done this, you can ask whether this functor is representable by a scheme or an ind-scheme (but I want to emphasize
that this question doesn't make sense before you define the functor).