Several questions on semi infinite flag manifold

Question

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!

Answer 1

Until you get a better answer this may help you. As far as I understand the semi-infinite flag manifold appeared in Feigin-Frenkel's paper [1] where they weren't really defined as an algebro-geometric object but rather they constructed what morally should be called some sheaves on them. Since both $G(K)$ and $N_-(K)\cdot H(\mathcal{O})$ are ind-groups, a priori their quotient as $k$-spaces is the sheafification of the quotient functor and it is not clear if it has a stratification by finite type schemes. I use the words finite type here because these spaces were introduced in the hopes that some category of perverse sheaves on them would be equivalent to some category of representations of the affine algebra for $g$ (generalizing Beilinson-Bernstein localization)(as Alexander Braverman noted below, we could deal with finite codimention strata as well). I quote from [2] (slightly different setup than yours)

...Since the pioneering work of Feigin and Frenkel [1], people were trying to develop the theory of perverse sheaves (constructible with respect to a given stratification) on G((t))/B((t)) ... The problem is that G((t))/B((t)) is very essentially infinite-dimensional, so that the conventional theory of perverse sheaves, defined for schemes of finite type, was not applicable. Since it was (and still is) not clear whether there exists a direct definition of perverse sheaves on G((t))/B((t))...

And similar quote from [3]
The semi-infinite flag manifold, thought of as $G(K )/N (K ) · T (\mathcal{O} )$, does not carry an algebro-geometric structure that would allow for the theory of perverse sheaves, or D-modules, in the way it is known today.

So it is my understanding that there has been work in trying to find inductive systems of schemes without success, in the case of the affine Grassmanian the situation is simplified because the $G(\mathcal{O})$ orbits are finite dimensional and there is a lattice model for the $GL(n)$ case. The general case and the affine flag case (not semi-infinite, but quotient by Iwahori) can be reduced to this $GL(n)$ case. In the semi-infinite situation, all orbits are infinite dimensional. The above mentioned article [3] in fact does construct a category that has all the properties that we would like perverse sheaves on $X$ to have, to that end the authors use an actual ind-scheme that serves to approximate $X$ (they use $Bun N$).

As for your second question regarding maps from a curve (the punctured disk) to the flag manifold, I'll refer you to section 3 and 4 chapter 1 (arxiv version) of [4] which by the way is the first attempt (as far as I know) of writing a category of perverse sheaves on these spaces. There's also a discussion there on the ind-scheme structures of related spaces constructed globally on a curve.

[1] B. Feigin, E. Frenkel, Affine Kac-Moody algebras and semi-infinite flag manifolds, Comm. Math. Phys. 128 (1990) 161–189.
[2] Braverman, A. Finkelberg, M.; Gaitsgory, D. Mirković, I. Iersection cohomology of Drinfeld's compactifications. Selecta Math. (N.S.) 8 (2002), no. 3, 381–418.
[3] Arkhipov, S. Braverman, A. Bezrukavnikov, R. Gaitsgory, D. Mirković, I. Modules over the small quantum group and semi-infinite flag manifold. Transform. Groups 10 (2005), no. 3-4, 279–362.
[4] Finkelberg, Michael; Mirković, Ivan Semi-infinite flags. I. Case of global curve P1. Differential topology, infinite-dimensional Lie algebras, and applications, 81–112, Amer. Math. Soc. Transl. Ser. 2, 194, Amer. Math. Soc., Providence, RI, 1999.

Answer 2

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).