Global Affine Flag Variety and Affine Flag Variety

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?

• One thing that's handy about it is that the family is topologically trivial, at least once one fattens $Fl$ to $LK/T$ and $Gr$ to $LK/K \cong \Omega K$, since $LK \cong K \times \Omega K$ gives $LK/T \cong K/T \times \Omega K$. – Allen Knutson Jan 24 '15 at 3:02
• Thanks! I know $Fl$ is topologically trivial, but I am not sure how to make use of that at the moment. As for your comment, what is $K$? Is $T$ the maximal torus in $G$? – Qiao Jan 24 '15 at 16:35
• $K$ is the maximal compact in $G$, e.g. $U(n)$ inside $GL(n)$. I'm being careless notationally using $T$ as the maximal (compact) torus in $K$, rather than (complex) in $G$. Because the family is topologically trivial, you can correspond the $T$-fixed points in the general and special fibers. Then $X$ and $X_{\epsilon\to 0}$ must induce the same elements of equivariant $K$-homology, i.e. must be writable as the same combination of $T$-fixed points with $frac(K_T)$-coefficients. – Allen Knutson Jan 26 '15 at 0:57
• Thanks! Is there any references for your comment? Why the family being topologically trivial imply that we could correspond the $T-$fixed points? In the example in my question, a copy of $\mathbb{P}^1$ degenerates to two copies of $\mathbb{P}^1$ glued at a point. – Qiao Jan 26 '15 at 17:40
• The family is (topologically) trivial; the subfamily is not. For a smaller example, imagine the curve $xz-t\ y^2$ in $\mathbb P^2$ with coordinates $x,y,z$ and parameter $t$. This is a nontrivial subfamily of the (algebraically!) trivial family over $Spec\ k[t]$ with constant fiber $\mathbb P^2$. When you say a family is trivial (in whatever sense), you mean you can correspond each fiber to some fixed fiber; I'm just saying that in the Gaitsgory example, that correspondence can be made $T$-equivariant. Therefore the (discrete set of) fixed points must correspond. – Allen Knutson Jan 27 '15 at 19:47

2 Answers

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.

Let $G$ be a split reductive group over a field $k$. Let $T\subset B$ a maximal split torus contained in a Borel. Denote by $X_*(T)$ the group of cocharacters of $T$. To every $\mu\in X_*(T)$ there is associated an element $t^\mu\in G(k((t)))$. Let $Gr_\mu$ be the reduced closure of the $G(k[[t]])$-orbit of $t^\mu\cdot e_0$ in $Gr$, where $e_0$ denotes the base point of $Gr$. In this case the image of the closed subvariety $Gr_\mu\times\{e\}\subset Gr\times G/B$ in the affine flag variety $Fl$ as $\epsilon\to 0$ was first determined by X. Zhu in his paper "On the coherence conjecture of Pappas and Rapoport", Theorem 3.8:

http://arxiv.org/pdf/1012.5979v3.pdf

Let me sketch his result. Let $I$ be the preimage of the Borel $B$ under the reduction map $G(k[[t]])\to G$, $t\mapsto 0$. For every $\mu\in X_*(T)$, let $Fl_\mu$ be the reduced closure of the $I$-orbit of $t^\mu\cdot e_0$ in $Fl$. Then Zhu shows that the image of $Gr_\mu\times\{e\}$ in $Fl$ is the union of the $Fl_\lambda$, where $\lambda$ runs over all translates of $\mu$ under the finite Weyl group acting on $X_*(T)$.

In the case $G=Gl_n$, $B$ the upper triangular Borel and $T$ the diagonal torus, we have $X_*(T)=\mathbb{Z}^n$ and the finite Weyl group is just the symmetric group on $n$ elements acting on $\mathbb{Z}^n$ by permuting the coordinates.

Note that Zhu's family $Fl_{\mathbb{A}^1}$ slightly differs from Gaitsgory's family: there is not extra $G/B$-factor. In fact his results work for more general reductive groups and use the theory of Bruhat-Tits group schemes.