There is a construction of a global affine flag variety over $\mathbb{A}^1$ (or another curve) $Fl_{\mathbb{A}_1}$ such that each fiber above $\epsilon \neq 0$ is isomorphic to a direct product of the affine Grassmannian $Gr$ with the ordinary flag variety $G/B$, $Gr \times G/B$. The fiber above $\epsilon = 0$ is the affine flag variety $Fl$. This first appeared in Gaitsgory's paper 'Construction of central elements in the affine Hecke algebra via nearby cycles' http://arxiv.org/abs/math/9912074
If we have some projective varieties in $Gr \times G/B$, how do we find out more about their images in the affine flag variety $Fl$, as $\epsilon \rightarrow 0$? More precisely in page 5, section 1.2.3 of the paper above, there is this example
where for $G = GL_2$, a family of $\mathbb{P}^1 \subset Gr$ degenerates to two copies of $\mathbb{P}^1$ glued at a point, in the affine flag variety $Fl$ for $GL_2$.
How do we verify this? Could we calculate things like this by some concrete methods?
Best Answer
Now let me attempt to give an answer myself.
There are very concrete descriptions of the fibers $Fl_{\epsilon}$ in $Fl_{\mathbb{A}^1}$ for each $\epsilon \in \mathbb{A}^1$.
$Fl_{\epsilon} \cong LG/I_{\epsilon}$, where $LG = G(k((t)))$ is the loop group of the algebraic group $G$, and $I_{\epsilon}$ is the pre-image of the Borel subgroup $B$ under the map $G(k((t))) \rightarrow G$ by evaluating at $t = \epsilon$.
There is also a lattice picture of $Fl_{\epsilon}$ for type A. $Fl_{\epsilon}$ is the moduli space of the following data: a lattice $L$ and a flag $f$ in the vector space $L/(t - \epsilon)L$. When $\epsilon = 0$, we recover the usual lattice picture of the affine flag variety $Fl$.
In Gaitsgory's example above, $G = GL_2$, $Y_0$ is the moduli space of lattices $L$ contained in $L^0 = \mathcal{O} \oplus \mathcal{O}$ with $\dim(L^0/L) = 1$. $Y_0$ is isomorphic to $\mathbb{P}^1$ and we are interested in the closure of its image in $Fl$ as $\epsilon \rightarrow 0$.
Let $a_1, a_2$ denote the two $T-$fixed points of $Y_0$ such that $a_1 = \mathcal{O} \oplus t\mathcal{O}$ and $a_2 = t\mathcal{O} \oplus \mathcal{O}$ as lattices. As $\epsilon \rightarrow 0$, the image of $a_1$ and $a_2$ are the points $(a_1, l), (a_2, l')$ in $Fl$, where $l$ and $l'$ are the lines fixed by $B$ in the respective flag varieties.
Another $T-$fixed point in $Fl$ that is in the closure of the image of $Y_0$ is $(a_1, l')$. For the family of lattices $a_1 + \epsilon^2 a_2$, $l'$ is the right flag to pick in $G/B$ $\forall \epsilon \neq 0$.
Overall, the image of $Y_0$ in $Fl$ as $\epsilon \rightarrow 0$ is two copies of $\mathbb{P}^1$ that connects these three $T-$fixed points.