Affine Grassmannians of reductive groups

This serves as a starting point for my self-study of geometric Langlands. Currently, I am interested in geometric Satake equivalence, which establishes an equivalence between the category of representation of the Langland dual group (of a given reductive group) and the category of perverse sheaves on affine Grassmannians. Affine grassmannians are interesting objects, and appear naturally when studying stack of principal bundles. Let us now study them, and the story, of reductive groups, is often with $\mathrm{GL}_n$. 

Affine Grassmannians of general linear groups via lattices

Let $R$ be a commutative ring and denoteby $R[[t]]$ and $R((t))$ the formal power series and Laurent series, respectively. 

Definition. A lattice $L \subset R((t))^n$ is a finite locally free $R[[t]]$-submodule of $R((t))^n$ such that the canonical morphism $L \otimes_{R[[t]]} R((t)) \longrightarrow R((t))^n$ is an isomorphism.

Note that locally free means that a vector bundle at the level of schemes. Spelling out the definition, this means that there exist $f_1,...,f_n \in L$ so that $L[f^{-1}_i]$ is a free $R[[t]][f^{-1}_i]$-module.  

We get a characteriziation of lattices

Lemma. Given a finitely generated $R[[t]]$-submodule $L$ of $R((t))^n$, then $L$ is a lattice if and only if there is some $N \in \mathbb{N}$ so that $t^NR[[t]]^n \subset L \subset t^{-N}R[[t]]^n$.

Proof. Observe that one can instead show that there are $N_1,N_2$ with $t^{N_2}R[[t]]^n \subset L \subset t^{-N_1}R[[t]]^n$. One bound  $L \subset t^{-N_1}R[[t]]^n$ is obvious. Indeed, by finitely generated property,

$$L = R[[t]]\left < l_1(t),...,l_m(t) \right> \subset R((t))^n.$$ Since each $l_i$ is of the form $t^{N_i}f_i(t)$ with $N_i \in \mathbb{Z},f_i(t) \in R[[t]]$, one can clear the denominator to obtain the bound. The other bound is less obvious. Suppose now that $t^{N}R[[t]]^n \subset L$, then by tensoring one gets

$$R((t))^n = t^NR[[t]]^n \otimes_{R[[t]]} R((t)) \subset L \otimes_{R[[t]]} R((t)) \subset R((t))^n$$ and hence equality holds. For the other direction, we still choose a presentation of $L$ as above, then the hypothesis $ L \otimes_{R[[t]]} R((t)) \longrightarrow R((t))^n$ implies that there are equations

$$ \begin{pmatrix}
 a_{11}&  . & a_{1r} \\
. & .   & . \\
 a_{r1}& .  & a_{rr} \\
\end{pmatrix}\begin{pmatrix}
l_1 \\. \\
l_r
\end{pmatrix} = \begin{pmatrix}e_1 \\ . \\
e_r
\end{pmatrix}$$ where $a_{ij} \in R((t))$ and $e_1,...,e_n \in R((t))^n$ are standard basis. By writing $a_{ij}= t^{N_{ij}}$ and clear the denominator again, one obtains the desired bound. 

From this, one can produce miltiple examples of lattices, we leave this to the reader. The next result shows that we can work locally on $R$ (in the definition, it is locally on $R[[t]]$). This is used in the proof of the ind-presentation of Affine Grassmannians below.

Lemma. Let $L \subset R((t))^n$ be lattice. There are $f_1,...,f_r \in R$ with $\left <f_1,...,f_r \right> = R$ and $L \otimes_{R[[t]]} R[f^{-1}_i][[t]]$ are free $R[f_i^{-1}][[t]]$-modules.

Proof. By definitions, there are $g_1,....,g_r \in R[[t]]$ with $\left <g_1,...,g_r \right> = R[[t]]$ and 

$$L[g_i^{-1}] = L \otimes_{R[[t]]} R[[t]][g_i^{-1}]$$ is a free $R[[t]][g_i^{-1}]$-module. We set $f_i = g_i(0)$ and by substituting $t=0$ into the equation $\left <g_1,...,g_r \right> = R[[t]]$ one gets that $\left <f_1,...,f_r \right> = R$. We observe that $g_i$ is invertible in $R[f_i^{-1}][[t]]$ and $R[[t]][g_i^{-1}] \subset R[f_i^{-1}][[t]]$ and hence

$$\begin{align*} L \otimes_{R[[t]]} R[f_i^{-1}][[t]] & = L \otimes_{R[[t]]} R[[t]][g_i^{-1}] \otimes_{R[[t]][g_i^{-1}]} R[f_i^{-1}][[t]] \\ & = (R[[t]][g_i^{-1}])^n \otimes_{R[[t]][g_i^{-1}]} R[f_i^{-1}][[t]] \\ &= R[f_i^{-1}][[t]] \end{align*}$$

as desired.

We get straight to the definition of affine Grassmannians.

Definition. The Affine Grassmannians of $\mathrm{GL}_n$ is the functor 

$$\begin{align*} \mathrm{Gr}_{\mathrm{GL}_n} \colon \mathbb{C}-\text{Alg} & \longrightarrow \textbf{Sets} \\ R & \longmapsto \left \{ \text{lattices of} \ R((t))^n \right \}. \end{align*}$$

The following theorem shows that Affine Grassmannians are ind-schemes, and in fact ind-projective schemes. Later, we generalize the definition to any group, and for reductive group, their Affine Grassmannians are ind-projective as well.

Theorem. The functor $ \mathrm{Gr}_{\mathrm{GL}_n}$ is an ind-projective ind-scheme, meaning that there are projective $\mathbb{C}$-scheme $\mathrm{Gr}_{\mathrm{GL}_n}^N$ together with closed immersions $\mathrm{Gr}_{\mathrm{GL}_n}^N \hookrightarrow \mathrm{Gr}_{\mathrm{GL}_n}^{N+1}$ and an isomorphism 

$$\mathrm{Gr}_{\mathrm{GL}_n}(R) \simeq \varinjlim_{N \to \infty} \mathrm{Gr}_{\mathrm{GL}_n}^N(R)$$ for any $\mathbb{C}$-algebra $R$. 

Let us reveal the definitions of these schemes $\mathrm{Gr}_{\mathrm{GL}_n}^N$. At the level of functors of points,

$$\mathrm{Gr}_{\mathrm{GL}_n}(R) = \left \{L \subset R((t))^n \ \text{a lattice} \mid t^NR[[t]]^n \subset L \subset t^{-N}R[[t]]^n \right \}.$$ The transitions $\mathrm{Gr}_{\mathrm{GL}_n}^N \hookrightarrow \mathrm{Gr}_{\mathrm{GL}_n}^{N+1}$ are obvious, sending a lattice to itself (it is not obvious that they are closed immersions). By the first lemma, it is clear that 

$$\mathrm{Gr}_{\mathrm{GL}_n}(R) \simeq \varinjlim_{N \to \infty} \mathrm{Gr}_{\mathrm{GL}_n}^N(R)$$ so the point is to show that $\mathrm{Gr}_{\mathrm{GL}_n}^N$'s are representable by projective schemes. Indeed, if they are represented by projective schemes, then transitions $\mathrm{Gr}_{\mathrm{GL}_n}^N \longrightarrow \mathrm{Gr}_{\mathrm{GL}_n}^{N+1}$ are proper as well, and they are monomorphisms so they must be closed immersions. 

Proof. We set

$$M_R = \frac{t^{-N}R[[t]]^n}{t^NR[[t]]^n} = \left \{\bullet t^{-N+1} + \cdots + \bullet t^{N-1}+\bullet t^N \mid \bullet \in R^n \right \} \simeq R^{2Nn}.$$ We define a map

$$\begin{align*} \mathrm{Gr}_{\mathrm{GL}_n}^N(R) & \longmapsto \left \{U \subset M_R \mid M_R/U \ \text{locally free R-module} \right \} \\ L & \longmapsto \frac{L}{t^NR[[t]]^n }. \end{align*}$$ To show that this map is well-define, one needs to check that 

$$\frac{M_R}{L/(t^NR[[t]]^n)} \simeq {t^{-N}R[[t]]^n}{L}$$ is a locally free $R$-module. By the second lemma, we can work locally on $R$. Thus, we can assume that there are $f_1,...,f_r \in R$ with $\left <f_1,...,f_r \right> = R$ and $L \otimes_{R[[t]]} R_{f_i}[[t]]$ are free $R_{f_i}[[t]]$-modules. Therefore, it is safe to assume that $L$ is free in which case we have

$$R((t))^n/L  = L[t^{-1}]/L \simeq R((t))^n/R[[t]]^n \simeq \bigoplus_{i<0}t^i R^n$$ and there is an exact sequence

$$0 \longrightarrow t^{-N}R[[t]]^n/L \longrightarrow R((t))^n/L \longrightarrow R((t))^n/t^{-N}R[[t]] \longrightarrow 0.$$ The term $R((t))^n/t^{-N}R[[t]]$ is free and hence projective, so the sequence splits which shows that the term $ t^{-N}R[[t]]^n/L$ is a direct summand of $R((t))^n/L$, which is free (hence projective), which forces itself to be projective. For finitely generated modules, being projective is equivalent to being locally free so we win, the map is well-defined. One sees that the functor 

$$R \longmapsto \left \{U \subset M_R \mid M_R/U \ \text{locally free R-module} \right \}$$ is nothing but the usual Grassmannians $\mathrm{Gr}(\bullet,2Nn)$ (with $\bullet = 2Nn  - \mathrm{rank}(L)$) hence projective. We are done if we can show that the map above identifies $ \mathrm{Gr}_{\mathrm{GL}_n}^N$ with a closed subscheme of $\mathrm{Gr}(\bullet,2Nn)$. Now on $M_R$ there is a nilpotent operation, namely, multiplying with $t$ ($t^{2N}=0$), we claim that $\mathrm{Gr}_{\mathrm{GL}_n}^N$ can be identified with the following functor

$$\mathrm{Gr}(\bullet,2Nn)^t(R) = \left \{ U \subset M_R \mid M_R/U \ \text{locally free and} \ tU \subset U \right \}$$ which is a subfunctor of $\mathrm{Gr}(\bullet,2Nn)$. We finish if we can prove the following two lemmas.

Lemma. $\mathrm{Gr}(\bullet,2Nn)^t \subset \mathrm{Gr}(\bullet,2Nn)$ is a closed subfunctor.

Proof. Indeed, using the basic $t^{-N+1}e_1,...,t^Ne_1,t^{-N+1}e_2,...$ we see that the operation $t$ has $n$ Jordan blocks of length $2N$. Covering $\mathrm{Gr}(\bullet,2Nn)$ by standard charts, we see that $\mathrm{Gr}(\bullet,2Nn)^t$ is defined by explicit equations.

Lemma. The map $ \mathrm{Gr}_{\mathrm{GL}_n}^N \longrightarrow \mathrm{Gr}(\bullet,2Nn)$ is an isomorphism. 

Proof. The map is clearly injective because $t^NR[[t]]^n \subset L$ by assumption. For surjectivity, we first reduce to the case that $R$ is noetherian. By the lemma above, $\mathrm{Gr}(\bullet,2Nn)^t$ is locally of finite type so 

$$\mathrm{Gr}(\bullet,2Nn)^t(\varinjlim R_i) = \varinjlim \mathrm{Gr}(\bullet,2Nn)^t(R_i).$$ Writing $R$ as a union of finitely generated $\mathbb{C}$-algebra, we can assume $R$ itself finitely generated over $\mathbb{C}$ and hence noetherian by Hilbert's basis theorem. If $R$ is noetherian, the inclusion $R[t] \longrightarrow R[[t]]$ is flat by this StackTag. We have a map

$$\varphi \colon t^{-N}R[t]^n \longrightarrow \frac{t^{-N}R[t]^n}{t^NR[t]^n} \simeq \frac{t^{-N}R[[t]]^n}{t^NR[[t]]^n} \longrightarrow t^{-N}R[[t]]^n/U$$ and its kernel $L = \operatorname{Ker}(\varphi)$. By flatness, 

$$(L \otimes_{R[t]} R[[t]])/t^NR[[t]]^n = U$$ so if we can show that $L$ is locally free $R[t]$-module, so is $U$. We may assume that $R=(R,\mathfrak{m},k)$ is a local ring. Using this StackTag, one can instead try to prove that $L/\mathfrak{m}L = L \otimes_R R/\mathfrak{m}$ is locally free $k[t]$-module. Now $M_R/U$ is $R$-flat so we see that $L/\mathfrak{m}L$ is a $(R/\mathfrak{m})[t]$-submodule of $t^{-N}(R/\mathfrak{m})[t]^n$ and therefore $t$-torsion free. However, $(R/\mathfrak{m})[t]$ is PID, which implies that $L/\mathfrak{m}L$ is flat as well, but for a finitely presented module, being flat (or projective) and being locally free are equivalent so we win.

Affine Grassmannians of general groups via torsors

To define Affine Grassmannians of general group, we need to change the definition as the notion of lattices does not work. We reformulate the definition in terms of $\mathrm{GL}_n$-torsors.

Lemma. The functor $\mathbb{C}-\text{Alg} \longrightarrow \textbf{Sets}$ sends $R$ to pairs $(\mathcal{E},\beta)$ with 

• $\mathcal{E}$ is a $\mathrm{GL}_n$-torsor over $\operatorname{Spec}(R[[t]])$.
• $\beta \colon \mathcal{E} \times_{R[[t]]} \operatorname{Spec}\big(R((t)) \big) \simeq \mathrm{GL}_n(R[[t]]) \times_{R[[t]]} \operatorname{Spec}\big(R((t)) \big)$ an isomorphism

is isomorphic to $\mathrm{Gr}_{\mathrm{GL}_n}$. 

Proof. This is obvious given the fact that any $\mathrm{GL}_n$-torsor is Zariski-locally trivial. 

Thus, the lemma suggests the following definitions.

Definition. Let $G/\mathbb{C}$ be an affine algebraic group, we define the Affine Grassmannian $\mathrm{Gr}_G$ as a functor $\mathbb{C}-\text{Alg} \longrightarrow \mathbf{Sets}$ that sends $R$ to pairs $(\mathcal{E},\beta)$:

• $\mathcal{E}$ is a $G_{R[[t]]}$-torsor on $\operatorname{Spec}(R[[t]])$.
• $\beta \colon \mathcal{E} \times_{R[[t]]} \operatorname{Spec}\big(R((t)) \big) \simeq G(R[[t]]) \times_{R[[t]]} \operatorname{Spec}\big(R((t)) \big)$ an isomorphism. 

Building upon the previous theorem, one has

Theorem. If $G$ is reductive, $\mathrm{Gr}_G$ is an ind-projective ind-scheme. 

Proof. Matsushima's theorem claims that if $G$ is reductive, then $H \subset G$ is reductive if and only if $G/H$ is affine. From this, we can find a closed immersion $G \subset \mathrm{GL}_n$ so that $\mathrm{GL}_n/G$ is affine. With this datum, we can show that $\mathrm{Gr}_G \longrightarrow \mathrm{Gr}_{\mathrm{GL}_n}$ is ind-closed. However, I do not know any elementary proof of this without using stacks and heave descent theory. Better to omit it.