\section{Introduction to Affine Grassmanians (Notes by Marin Petkovic)}

Let $k$ be a field and let $G$ be a split connected reductive group over $k$. Let $F=k((t))$ and $\mc O=k[[t]]$.
The aim of this section is to define the affine Grassmanian of $G$ as a geometric object. That is, we want to define $\gr{G}$ so that $\gr{G}(k)={G(F)}/{G(\mc O})$. Before defining the Grassmanian for general groups, we are going to consider the special case of $\mathrm{GL}_n$. 


\subsection{The affine Grassmannian of $\mathrm{GL}_n$}





Recall that by the Cartan decomposition
$$\gl{n}{F}=\coprod_{\lambda\in X_\bullet(T)_+} \gl{n}{\mc O}\lambda(t)\gl{n}{\mc O}.$$
We proved this using the lattice $\Lambda_0=\bigoplus_{i=1}^n k[[t]]e_i\subset \bigoplus_{i=1}^n k((t))e_i$.
Consider the group action
\begin{align*}\gl{n}{F}&\longrightarrow \{ \mc O \text{-lattices in } V\}\\
g&\longmapsto g\cdot \Lambda_0.
\end{align*}
Then $\Stab(\Lambda_0)=\mathrm{GL}_n(\mc O)$ and we get a bijection
$$\quotient{\gl{n}{F}}{\gl{n}{\mc O}}\longiso \{ \mc O \text{-lattices in } V\}.$$
This motivates the following definition.

%\begin{defn}
%The affine Grassmannian of the group $\text{GL}_n$ is the functor $$\gr{\mathrm{GL}_n}:k\text{-alg}\rightarrow \text{Set}$$ given by
%$$R\longmapsto \{R\text{-families of lattices in }k((t))^n\},$$
%where an $R$-family of lattices in $k((t))^n$ is a finitely generated projective $R[[t]]$-module $\Lambda\subset R((t))^n$ such that $\Lambda\otimes_{R[[t]]} R((t))=R((t))^n$.
%\end{defn}
\begin{defn}
The affine Grassmannian of the group $\text{GL}_n$ is the functor 
\begin{align*}
\gr{\mathrm{GL}_n}:k\text{-alg}&\longrightarrow \text{Set}
\end{align*}
given by
\begin{align*}
R&\longmapsto \left\{
\begin{array}{c}  
\text{ f. g. projective } R[[t]] \text{-submodules }\Lambda\subset k((t))^n \\ \text{ such that }\Lambda\otimes_{R[[t]]} R((t))=R((t))^n
 \end{array}
\right\}
\end{align*}

\end{defn}

\begin{defn}
A $k$-space is a functor $\mc F:k\text{-alg}\rightarrow \text{Set}$ that is a sheaf in fpqc topology, i.e.
\begin{enumerate}[(1)]
\item $\mc F$ is a sheaf in the Zariski topology, and
\item for all homomorphisms $R\rightarrow R'$ faithfully flat, the sequence
$$\mc F(R)\longrightarrow \mc F(R') \begin{array}{c}
\displaystyle \overset{p_1^\ast}{\longrightarrow} \\[-5pt]
\displaystyle\underset{p_2^\ast}{\longrightarrow}
\end{array} \mc F(R'\otimes_R R')$$
is exact.
\end{enumerate}
\end{defn}
\begin{rmk}
Every scheme over $k$ is a $k$-space. (fpqc descent)
\end{rmk}
\begin{defn}
An ind-scheme is a directed system $(X_i)_{i\in I}$ of schemes such that all transition maps $X_i\rightarrow X_j$ are closed immersions. We say that an ind-scheme is ind-projective, if $X_i$'s can be chosen to be projective. Similarly we define other properties of ind-schemes. 
\end{defn}
\begin{exmpl}
An example of ind-scheme is $$\mathbb P_\infty=\lim_{\longrightarrow} \mathbb P_n.$$
\end{exmpl}
It will turn out that $\gr{\mathrm{GL}_n}$ is not representable by a scheme, but it is an ind-scheme.
\begin{thm}
The affine Grassmanian $\gr{\mathrm{GL}_n}$ is an ind-projective ind-scheme.
\end{thm}
\begin{proof}
We can write $\gr{\mathrm{GL}_n}=\bigcup_{N\geq 1}\mathrm{Gr}^{(N)}$ where $\mathrm{Gr}^{(N)}$ is a subfunctor defined as
$$\mathrm{Gr}^{(N)}:R\longrightarrow\{\Lambda\in \gr{\mathrm{GL}_n}(R)\;|\;t^NR[[t]]^n\subset \Lambda\subset t^{-N}R[[t]]^n\}$$
Define functors $Q^{(N)}$ with
$$Q^{(N)}(R)=\left\{
\begin{array}{c} R[[t]] \text{-quotients }\bar\Lambda \text{ of } \quotient{t^{-N}R[[t]]^n}{t^{N}R[[t]]^n}\\
\text{ such that } \bar\Lambda \text{ is a projective } R\text{-module}
\end{array}
\right\}$$
and a morphism of functors $\mathrm{Gr}^{(N)}\longrightarrow Q^{(N)}$ defined with $$\Lambda\longmapsto \bar\Lambda=\quotient{ t^{-N}R[[t]]^n}{\Lambda}.$$
We claim that this is an isomorphism of functors. Granted this, we are done, because
$$Q^{(N)}\xhookrightarrow{}\mathrm{Grass}\left(\quotient{t^{-N}k[[t]]^n}{t^Nk[[t]]^n}\right)$$
identifies $Q^{(N)}$ as a closed subscheme of $t$-stable quotients of $\quotient{t^{-N}k[[t]]^n}{t^Nk[[t]]^n}$.




To prove the claim, we first check that $t^{-N}R[[t]]^n/\Lambda$ is a  projective $R$-module. For that, consider the short exact sequence
$$0\longrightarrow \quotient{t^{-N}R[[t]]^n}{\Lambda}\longrightarrow 
\quotient{R((t))^n}{\Lambda}\longrightarrow \quotient{R((t))^n}{t^{-N}R[[t]]^n}\longrightarrow 0.$$
Since $\Lambda$ is projective, so is $\Lambda/t\Lambda$.
$$\quotient{R((t))^n}{\Lambda}\iso \bigoplus \quotient{t^{-k}\Lambda}{t^{-k+1}\Lambda}$$
is projective since each of the summands on the right is isomorphic to $\Lambda/t\Lambda$ as $R$-modules. The same reasoning shows that $\quotient{R((t))^n}{t^{-N}R[[t]]^n}$ is projective. From the short exact sequence above it follows that $t^{-N}R[[t]]^n/\Lambda$ is projective.



To show bijectivity, let $Q\in Q^{(N)}(R)$. As we have seen, $Q^{(N)}$ is of finite type over $k$, so for all $k$-algebras $R$ we have $$Q^{(N)}(R)=\underset{\text{f.g. }R_i\subset R}{\text{colim}}Q^{(N)}(R_i).$$
We may therefore assume $R$ is finitely-generated over $k$.

We have 
$$t^{-N}R[[t]]^n\rsurj \quotient{t^{-N}R[[t]]^n}{t^{N}R[[t]]^n}\rsurj Q,$$
so let $\Lambda_Q$ be the kernel of the  composition above. We need to show that $\Lambda_Q\in \mathrm{Gr}^{(N)}{(R)}$. Since $t^NR[[t]]^n\subset\Lambda_Q$, we know that $\Lambda\otimes R((t))=R((t))^n$, so it remains to show that $\Lambda_Q$ is finitely generated projective $R[[t]]$-module.
Since $R$ is finitely generated over $k$, it is Noetherian. In particular $R[t]\longrightarrow R[[t]]$ is flat.

Since $$t^{-N}R[t]^n\rsurj\quotient{t^{-N}R[t]^n}{t^{N}R[t]^n}\isomorphism\quotient{t^{-N}R[[t]]^n}{t^{N}R[[t]]^n}\rsurj Q$$
we can define $\Lambda_{Q,\text{fin}}=\ker(t^{-N}R[t]^n\rsurj Q)$. Since $R[t]\longrightarrow R[[t]]$ is flat, $\Lambda_{Q,\text{fin}}\otimes_{R[t]} R[[t]]=\Lambda_Q$. Hence it suffices to show that $\Lambda_{Q,\text{fin}}$ is finitely generated projective $R[t]$-module. It is clearly finitely generated, since it is contained in $t^{-N}R[t]^n$.
To show $\Lambda_{Q,\text{fin}}$ is projective, we will show it is flat over $R[t]$.

For this, notice that $\forall \mf p\in \spec R$, $\Lambda_{Q,\text{fin}}\otimes_{R[t]} \kappa(\mf p)[t]$ is flat (and therefore free) over $\kappa(\mf p)[t]$. Indeed, since $Q$ is $R$-flat, we have
$$\Lambda_{Q,\text{fin}}\otimes_{R[t]} \kappa(\mf p)[t]=\ker\left(t^{-N}\kappa(\mf p)[t]^n\longrightarrow Q\otimes_R \kappa(\mf p)\right).$$
Since $t^{-N}\kappa(\mf p)[t]^n$ is torsion free, so is $\Lambda_{Q,\text{fin}}\otimes_{R[t]} \kappa(\mf p)[t]$, and hence it is flat.
Now apply the following lemma on $R_\mf p\longrightarrow R[t]_{\mf q}$ and $M=\Lambda_{Q,\text{fin}}\otimes R[t]_\mf q$, for all primes $\mf q$ of $R[t]$.

\end{proof}
\begin{lem}
Let $R\longrightarrow S$ be a local homomorphism of Noetherian local rings, $\mf m\subset R$ maximal ideal and $M$ a finite $S$-module. If $M$ is flat over $R$ and $M/\mf mM$ is free over $S/\mf mS$, then $M$ is free over $S$ and $S$ is flat over $R$.
\end{lem}
\begin{proof}
Stacks Tag 00MH
\end{proof}

\subsection{Affine Grassmanian for general groups}
%Now we define $\gr{G}$ for general groups $G$.
Let $\mc D_R=\spec{R[[t]]}$ (family of  disks parametrized by $R$) and $\mc D_R^\ast=\spec{R((t))}$ (family of punctured disks parametrized by $R$).

We can identify
$$\gr{\mathrm{GL}_n}(R)=\quotient{\left\{(\mc E,\beta) \;\Big|\; 
\begin{array}{c}  \mc E \text{ is a vector bundle of rank } n \text{ on } \mc D_R \\ \text{ and } \beta:{\restr{\mc E}{\mc D_R^\ast} }\longiso{\text{ the trivial bundle on } \mc D_R^\ast} \end{array}
\right\}}{\iso}.$$

\begin{prop}
Let $G$ be a linear group over $k$ and $S$ quasi-compact scheme. The following categories are equivalent:
\begin{enumerate}[(1)]
\item fpqc sheafs $\mc P$ on $\mathrm{Aff}/S$ which are right $G$-torsors, i.e. $\mc P$ has a right $G$ action such that 
\begin{align*}
\mc P\times G&\longrightarrow \mc P\times_S\mc P\\
(s,g)&\longmapsto (s,sg)
\end{align*}
is an isomorphism, and for which there exists an fpqc cover $S'\longrightarrow S$ such that $\mc P(S')\neq \emptyset$.
\item scheme maps $\pi:\mc E\longrightarrow S$ with a right $G$-action on $\mc E$ equivariant for $\pi$ such that there exists a faithfully flat cover $S'\longrightarrow S$ such that $\mc E\times_S S' \simeq G\times S'$ as $G$-torsors.
\item faithfully flat maps $\mc E\longrightarrow S$ with a right $G$-action such that $\mc E\times_S  \mc E \simeq G\times \mc{E}$ as $G$-torsors.
\item If $G$ is smooth, as in (2) but require $S'\longrightarrow S$ \'etale.
\end{enumerate}
\end{prop}


\begin{rmk} All the non-obvious implications follow from faithfully flat descent theory.
\end{rmk}
\begin{defn}
A $G$-bundle is an object in any of the four equivalent categories above (we will use ($4$)). \textit{The} trivial $G$-bundle over $S$ is $\mc E^0=G\times_k S$ with $G$ acting on itself by right multiplication.

\end{defn}

\begin{defn}
The affine Grassmannian of $G$ is the functor
$$\gr{G}:k\text{-alg}\longrightarrow \mathrm{Set}$$
given by
$$R\longmapsto \quotient{\left\{(\mc E,\beta) \;\Big|\; 
\begin{array}{c}  \mc E \text{ is a } G \text{-bundle on }   \mc D_R \text{ and } \\  \beta:{\restr{\mc E}{\mc D_R^\ast} }\longiso{\mc E^0} \text{ is an isomorphism of } G\text{-bundles}\end{array}
\right\}}{\iso}.$$
\end{defn}


\textbf{Relation to previous definition for $\mathrm{GL}_n$}

For any $S$ there is an equivalence:
\begin{align*}
\left\{ \mathrm{GL}_n\text{-bundles over } S\right\} &\longrightarrow \left\{ \text{rank }n\text{ vector bundles over }S\right\}\\
(\mc E\rightarrow S)&\longmapsto \mc E \overset{\mathrm{GL}_n}{\times} \mathbb A^n
\end{align*}
where $\mc E {}^{\phantom{1}\mathrm{GL}_n}\!\times \mathbb A^n={\raisebox{-.2em}{$\mathrm{GL}_n$} \left\backslash \raisebox{.2em}{$\mc E \times \mathbb A^n$}\right.}$ with the action given by $g.(s,v)=(sg^{-1},gv)$.
Its quasi-inverse is given by
$$\mc V\longmapsto \left( T\mapsto \underset{{}_{\mathrm{GL}_n}\!\circlearrowleft}{\mathrm{Isom}_T(\mc O_T^n, \mc V)}\right),$$
for all $\mc V$ rank $n$ vector bundle over $S$.

\begin{thm}
For any affine $G/k$ the presheaf $\gr{G}$ is representable by an ind-scheme of ind-finite type. If $G$ is reductive, then $\gr{G}$ is moreover ind-projective.
\end{thm}
\begin{proof}[Strategy of proof]
Choose an embedding $G\longrightarrow \mathrm{GL}_n$ such that $\mathrm{GL}_n/G$ is quasi-affine. In this case, one can show that the map 
\begin{align*}
\gr{G}&\longrightarrow\gr{\mathrm{GL}_n}\\
\mc E &\longmapsto \mc E \overset{G}{\times} \mathrm{GL}_n
\end{align*}
is a locally closed embedding.


When $G$ is reductive, $\mathrm{GL}_n/G$ is affine and the map above is a closed embedding.
\end{proof}
For general homomorphisms $H\longrightarrow G$, the map $\gr{H}\longrightarrow\gr{G}$ may be very strange.
\begin{exmpl}
Let $B<G$ be a Borel subgroup. If $G$ is reductive, the Iwasawa decomposition says $G(k((t)))=B(k((t)))\cdot G(k[[t]])$, so $\gr{B}\longrightarrow \gr{G}$ is a bijection on $k$-points. This map is far from being an isomorphism.
\end{exmpl}

\subsection{Another description of $\gr{G}$}
We would like a description of $\mathrm{Gr}_G$ that is closer to our intuitive understanding of it as the ``quotient'' $G(k((t)))/G(k[[t]])$. Define the loop group of $G$ to be
\begin{align*}
\mathrm{L}G:k\text{-alg}&\longrightarrow \text{Grp}\\
R&\longmapsto G(R((t)))
\end{align*}
and the positive loop group
\begin{align*}
\mathrm{L}^+G:k\text{-alg}&\longrightarrow \text{Grp}\\
R&\longmapsto G(R[[t]])
\end{align*}


\begin{lem} $\mathrm{L}^+G$ is a scheme over $k$ and $\mathrm L G$ is an ind-scheme over $k$.
\end{lem}
\begin{proof}[Proof for $\mathrm{GL}_n$]
Notice that
$$\mathrm L^+G(R)=\gl{n}{R[[t]]}=\left\{\left(\sum_{m\geq 0} b_{i,j,m}t^m\right)_{i,j}\Big| \; b_{i,j,m}\in R,\; \det\in R^\times\right\}$$
can be embedded into the affine scheme $R[\{x_{i,j,m}\},\det]$, and

$$\mathrm LG(R)=\bigcup_{r\geq 0}\{\text{matrices with all entries having poles of order at most } r\}$$
and all the objects on the right are schemes.
\end{proof}


Now consider the presheaf 
$$\quotient{\mathrm LG}{\mathrm L^+G}:R\longmapsto \quotient{\mathrm LG(R)}{\mathrm L^+G(R)}.$$
Let $\left[\quotient{\mathrm LG}{\mathrm L^+G}\right]$ be the sheafication for the fpqc (or fppf, or \'etale) topology. As we will see, when $G$ is smooth the choice of topology does not matter. 


\begin{prop}For any smooth linear algebraic group $G/k$, $\gr{G}$ is naturally isomorphic to $\left[\quotient{\mathrm LG}{\mathrm L^+G}\right]$.
\end{prop}
Before the proof, we apply the Proposition to an example:
\begin{exmpl}
Consider $\mathrm L\gm/\mathrm L^+\gm$. Let $f=\sum a_nt^n\in \mathrm L\gm(R)=R((t))^\times$. Then there exists $n_0$ such that $a_{n_0}\in R^\times$ and $a_n$ is nilpotent for all $n<n_0$ 
(eg. if $R=k[\epsilon]/\epsilon^2$, then $\epsilon t^{-1}+1\in R((t))^\times$ since $(\epsilon t^{-1}+1)(-\epsilon t^{-1}+1)=1$). Write $f(t)=a_{n_0}t^{n_0}g(t)$ where the negative powers of $t$ in $g(t)$ have nilpotent coefficients.
Then $g(t)$ can be factored uniquely as a product of a polynomial with degrees $\leq 0$ and nilpotent coefficients with a polynomial in $1+tR[[t]]$.
Hence we obtain a decomposition
$$\mathrm L\gm=\gm\times \ints\times(R\mapsto 1+tR[[t]])\times(\text{some highly non-reduced ind-scheme}).$$
Under this composition $\mathrm L^+\gm=\gm\times \mathbb A^\infty$, so 
$$\quotient{\mathrm L\gm}{\mathrm L^+\gm}\iso \ints\times(\text{some highly non-reduced ind-scheme}).$$
In particular, $\gr{\gm}$ is not reduced and $\pi_0\gr{\gm}=\ints$.
\end{exmpl}

\begin{proof}[Proof of proposition]


There is an isomorphism
\begin{align*}
\mathrm LG(R)&\longiso \quotient{\left\{(\mc E,\beta,\epsilon)\;\Bigg|\;\begin{array}{c}
\mc E \text{ is a } G\text{-bundle on } \mc D_R\\
\beta:\restr{\mc E}{\mc D_R^\ast} \longiso{\mc E^0} \\
\epsilon:\mc E\longiso \mc E^0
\end{array}\right\}}{\iso}\\
g&\longmapsto (\mc E^0,g,\mathrm{id})
\end{align*}
where we interpret $g$ as an automorphism of $\mc E^0$ over $\mc D_R^\ast$.
Let $\mathrm LG\longrightarrow \gr{G}$ be the obvious morphism that forgets $\epsilon$. Since $\mathrm L^+G(R)=\aut(\mc E^0/\mc D_R)$, this map factors and we obtain $\quotient{\mathrm LG}{\mathrm L^+G}\longrightarrow \gr{G}$. Since $\gr{G}$ is a sheaf (for fpqc topology and hence for \'etale topology) this induces a map of sheaves
$$\left[\quotient{\mathrm LG}{\mathrm L^+G}\right]\longrightarrow \gr{G}.$$
It is easy to see that this is injective, so it remains to prove surjectivity. This follows from the following lemma.
\end{proof}


\begin{lem}
Any $G$-bundle $\mc E$ on $\mc D_R$ can be trivialized \'etale locally on $R$, i.e. there exists \'etale cover $R\longrightarrow R'$ such that $\mc E\otimes_{R[[t]]} R'[[t]]$ is isomorphic to the trivial bundle on $\mc{D}_{R'}$.
\end{lem}
\begin{proof}
We claim that $\mc E$ is trivial over $\mc D_R$ iff $\mc E\times_{R[[t]]} R$ is trivial over $R$.
To see this, recall that $\restr{\mc E}{S'}$ is trivial iff $\mc E(S')\neq \emptyset$.
Since $\mc E$ is smooth, the maps $\mc E(R[t]/t^{n+1})\longrightarrow \mc E(R[t]/t^n)$ are all surjections.
Since $\mc E$ is affine, $\mc E(R[[t]])=\underset{\leftarrow}{\lim}\mc E(R[t]/t^n)$.
Thus, if $(\mc E\times_{R[[t]]} R)(R)\neq 0$, then $\mc E(R[[t]])\neq 0$, which is equivalent to $\mc E$ being trivial over $\mc D_R$.

Having established the claim, the lemma follows since $\mc E\otimes_{R[[t]]} R$ is a $G$-bundle over $R$, so it can be trivialized \'etale locally on $R$.
\end{proof}



Let $G$ be connected reductive group and $T\subset B\subset G$ a maximal torus and Borel subgroup. Let $\lambda\in X_\bullet(T)_+$. Then $\lambda(t)\in \gr{G}(k)$. The positive loop group $\mathrm L^+ G$ acts on $\gr{G}=\left[\quotient{\mathrm LG}{\mathrm L^+G}\right]$ by left multiplication.
\begin{defn}
Let $\gr{G,\lambda}$ be the $\mathrm L^+G$-orbit of $\lambda(t)$, regarded as a locally closed reduced subscheme of $\gr{G}$. Define $\gr{G,\leq \lambda}=\overline{\gr{G,\lambda}}\hookrightarrow \gr{G}$.
\end{defn}


Now we can answer the motivating question from classical Satake:

There is a complex of sheaves $IC_\lambda$ called the intersection complex on $\gr{G,\leq\lambda}$ such that under the sheaf-function dictionary:

\begin{align*}
\mc H_{G(k[[t]])} &\longiso R(G^{\vee})\\
\tr(IC_\lambda)&\longmapsto ch(V(\lambda)).
\end{align*}

The intersection complex is the basic example of a perverse sheaf; we turn to this subject in the next part of the course.
