Denote semi infinite flag manifold by $Fl_{\infty/2}=G((t))/N_-((t))H[[t]]$, denote $B_-((t))=N_-((t))H[[t]]$

from the book of Frenkel and Benzvi" Vertex algebras and algebraic curves", They take group $\hat{N_+}$ which is Lie group corresponding to unipotent radical of affine Kac-Moody algebra $\hat{g}$ respect to Standard Cartan decomposition of $\hat{g}$. Then they consider the orbit of action of $\hat{N_+}$ on $Fl_{\infty/2}$ and the $\hat{N_+}$-orbit of unit coset is isomorphic to $N_+[[t]]$ which is subspace of big cell ($N_+((t))$) of $Fl_{\infty/2}$. My question is:

What are the other $\hat{N_+}$-orbit of $Fl_{\infty/2}$? Is there any stratification of $Fl_{\infty/2}$ into these orbits?

Roman Fedorov told me that it may make more sense to consider $B_-((t))($ action on $G((t))/\hat{N_+}$. Then it seems similar to what Gaitsgory ever considered in his paper "twisted Whittaker model..." He considered the action of $N_-((t))$ onto affine Grassmannian $G((t))/G[[t]]$. And he claimed that the orbits are essentially infinite dimensional. But he did not write down the explicit decomposition.

So my question is how to write down this decomposition and how to write down the orbit decompostion of $\hat{N_+}$ acting on $Fl_{\infty/2}$?