From series papers of Frenkel-Feign-Gaitsgory-Braveman and Beilinson Drinfeld and their school we know that the algebraic geometric definition and theory of semi infinite flag manifold were not strictly built. However, one still have many ingredients which imply it does exists. Here are some questions:
I denote semi infinite flag manifold by $Fl_{\infty/2}:=G((t))/N_-((t))H[[t]]$. One can know that the big cell of this flag is $N_+((t))$, and the algebra of regular functions on it can be realized as $M_{\lambda(t)}^*$ (In Kac's terminology, it is called "imaginary Verma module"). It is constructed as coindcution from one dimensional representation of $\mathfrak{h}((t))$:$\lambda(t):\mathfrak{h}((t))\rightarrow \mathbb{C}$.
Question: How to construct this module $M_{\lambda(t)}^*$ via localization at "homogeneous coordinate ring" of $Fl_{\infty/2}$?
Considering the finite dimensional case or even the flag manifold of symmetrizable Kac-Moody algebra corresponding to standard Cartan decomposition. We have the following construction:
$M(0)^*=(\oplus_{\lambda\in P_+}L_\lambda)$ localized at $(K_\lambda,\lambda\in P_+)^{-1}$
Where $M(0)^*$ is contragredient Verma module with 0 character. $\oplus_{\lambda\in P_+}L_\lambda$ is direct sum of integrable highest weight irreducible representations of symmetrizable Kac-Moody algebra $\hat{g}$ which can be seen as homogeneous coordinate ring of flag manifold of $\hat{g}$, where the multiplication of this ring is given by Cartan multiplication.
$K_\lambda$ form a multiplicative set (for symmetrizable Kac-Moody algebra,one can find this construction at Kashiwara's paper, for general Kac-Moody case, one can find similar construction in O.Mathieu's paper).
From the categorical geometry view point, if you take Proj-category: $Proj\oplus_{\lambda\in P_+}L_\lambda$. It should be viewed as category of quasi coherent sheaves on flag manifold, $Qcoh(G/B)$ (A.Joseph-Rosenberg-Lunts-Tanisaki adopt this view point to establish the quantized Belinson Bernstein localization for quantum group)
From the book of Frenkel-Benzvi,or just look at the big cell of $Fl_{\infty/2}$ which is $N_+((t))$, it is a ind-projective limit of affine scheme $N_+$. So one can never expect $Fl_{\infty/2}$ to be a scheme or ind-scheme. However, I think there still should be similar categorical construction for $Qcoh(Fl_{\infty/2})$ and similar localization functor to get $M_{\lambda(t)}^*$
There are some technique difficulty to mimic the construction in usual affine flag manifold. Because $Fl_{\infty/2}$ came from non standard Cartan decomposition of Kac-Moody algebra, $M_{\lambda(t)}^*$ is not in category O, so I did not know whether it still has irreducible quotient $L_\lambda$ which will be used in "localization construction".
I realized one can never expect satisfied answers, so any helpful comments and ideas are welcome, Thank you!
