\section{Statement of classical Satake isomorphism}
\begin{center}
Shiang Tang
\end{center}

Recall that if $K$ is a compact open subgroup of $G$ that is locally profinite, the set of isomorphism classes of smooth irreducible $G$-representations $\pi$ such that $\pi^{K}\neq 0$ is equivalent to the set of isomorphism classes of irreducible $\mathcal{H}_K$-modules $\pi^{K}$. The case of basic interest to us is the following situation: $F$ is a local field with ring of integers $\mathcal O$ and a uniformizer $\varpi$. $\mathbf{G}$ is a split connected reductive group scheme defined over $\mathcal O$, $G=\mathbf{G}(F)$, $K=\mathbf{G}(\mathcal O)$. In this case, $\mathcal H_K$ can be explicitly described.

The following theorem gives us the first information about $\mathcal H_K$.

\begin{thm}[Cartan decomposition]
For $G,K$ as above, 
$$G=\bigsqcup_{\mathrm{\lambda\in X_*(T)/W}} K\lambda(\varpi)K$$ (If we choose $B\supset T$, then we get $X_*(T)/W=X_*(T)_{+}$.)
\end{thm}

\begin{proof}[Proof for $GL_n$]
It uses the structure theory of modules over PID. Let $V=F^n \supset \Lambda = \mathcal O^n=\oplus_{i=1}^n e_i$. For $g \in GL_n(F)$, consider the translated lattice $g(\Lambda)$, these two lattices are commensurable so $\exists N$, such that $\varpi^N g(\Lambda) \subset \Lambda$. By theory of finite-generated $\mathcal O$-modules, $\exists$ a basis $f_1,\cdots, f_n$ of $\Lambda$ and integers $a_1\geq \cdots \geq a_n$ such that $\varpi^{a_1}f_1,\cdots,\varpi^{a_n}f_n$ form a basis of $\varpi^N g(\Lambda)$ and hence $\varpi^{a_1-N}f_1, \cdots, \varpi^{a_n-N}f_n$ form a basis of $g(\Lambda)$. Let $B$ be the matrix of $g$ under the basis $f_1,\cdots, f_n$ of $\Lambda$, so $g=SBS^{-1}$ for some $S \in GL_n(\mathcal O)$. Since both of $gf_1,\cdots,gf_n$ and $\varpi^{a_1-N}f_1, \cdots, \varpi^{a_n-N}f_n$ are $\mathcal{O}$-bases of $g(\Lambda)$, there exists $C\in GL_n(\mathcal O)$ such that $B=diag(\varpi^{a_1-N},\cdots,\varpi^{a_n-N})\cdot C$ and so $g=S\cdot diag(\varpi^{a_1-N},\cdots,\varpi^{a_n-N})\cdot CS^{-1} \in Kdiag(\varpi^{a_1-N},\cdots,\varpi^{a_n-N})K$. 
\end{proof}

\begin{cor}
$\mathcal H_K$ has a basis given by $\{\mathbf{1}_{K\lambda(\varpi)K}\}_{\lambda \in X_*(T)_{+}}$.
\end{cor}

\begin{thm}[Satake Isomorphism]
There is an \textbf{algebra isomorphism}
$$\mathcal H_K \to \mathcal H_{T(\mathcal O)}^W$$ where 
$\mathcal H_{T(\mathcal O)}=\mathcal H(T(F),T(\mathcal O))$, given by 
$$f \mapsto (\mathcal Sf)(t)=\delta(t)^{1/2}\int_Nf(tn)dn$$
where $\delta(t)=|\det(Ad(t)|_{\mathfrak n})|^{-1}$.
\end{thm}

We make the following \textbf{important remarks}.

\begin{rmk}
RHS=$H_{T(\mathcal O)}^W$. 
$$\mathbb C[X_*(T)] \cong C_c^{\infty}(T(F)/T(\mathcal O))$$
$$[\mu] \mapsto \mathbf 1_{\mu(\varpi)T(\mathcal O)}$$ is an isomorphism of algebras. So 
$$\mathcal H_K \to \mathcal H_{T(\mathcal O)}^W\cong \mathbb C[X_*(T)]^W=\mathbb C[X^*(T^{\vee})]^W=R(G^{\vee})\otimes \mathbb C$$
The isomorphism $\mathcal H_K \cong R(G^{\vee})\otimes \mathbb C$ is what geometric Satake will categorify. 
\end{rmk}

\begin{rmk}
$\mathcal H_K$ is commutative!
\end{rmk}

\begin{rmk}
Since $\mathcal H_K=H_{T(\mathcal O)}^W$ is commutative, the following sets are in bijections:
$$\{\textrm{simple $\mathcal H_K$-modules}\}\leftrightarrow\{\textrm{1-dimensional representations of $H_{T(\mathcal O)}^W$}\}\leftrightarrow$$

$$\{\textrm{$\mathbb C$-algebra homomorphisms: $H_{T(\mathcal O)}^W=\mathbb C[X^*(T^{\vee})]^W=\mathcal O_{T^{\vee}/W}\to \mathbb C$}\}\leftrightarrow(T^{\vee}/W)(\mathbb C)$$

$$\leftrightarrow\{\textrm{semisimple conjugacy classes in $G^{\vee}$}\}$$
The last bijection holds since every semisimple element of $G^{\vee}$ is contained in some maximal torus, and all maximal tori are $G^\vee$-conjugate.
\end{rmk}

The point of Remark 1.6 is that it induces the unramified case of the local Langlands program, on which we give a short introduction:
Let $\Pi(G)$ be the set of irreducible smooth admissible $G$-representations up to isomorphisms (the "automorphic side"). Let $\Phi(G)$ be the set of $L$-parameters for $G$ up to equivalence (the "Galois side"). To define $L$-parameter, we start with the definition of Weil group:

\begin{defn}[Weil group]
Given a non-archimedean local field $F$, we have the following exact sequence
$$1\to I_F \to \mathrm{Gal}(\bar F/F) \to \mathrm{Gal}(\bar k_F/k_F) \to 1$$ where $k_F$ is the residue field of $F$ and $\mathrm{Gal}(\bar k_F/k_F) \cong \hat{\ints}$ is generated by the Frobenius map $\mathrm{Frob}: x \mapsto x^{|k_F|}$, $I_F$ is the inertia group. Define $W_F$ to be the preimage of $\ints = <\mathrm{Frob}>$, so it fits into an exact sequence
$$1\to I_F \to W_F \to \ints \to 0$$
This defines $W_F$ as a group. As a topological group, we declare $I_F$ to be an open subgroup with its natural profinite topology. Note that $W_F$ is not a profinite group. 
\end{defn}

\begin{defn}
An $L$-parameter for $G$ is a pair $(r,N)$ where $$r: W_F \to G^{\vee}(\mathbb C)$$ is a continuous homomorphism and where $N$ is a nilpotent element in $Lie(G)$ such that
$$r(g)Nr(g)^{-1}=q^mN$$ where the image of $g$ in $\mathrm{Gal}(\bar k_F/k_F)$ is $\mathrm{Frob}^m$. 

Now we define $\Phi(G)$ to be $$\{(r,N)|r(\mathrm{Frob}) \textrm{ is semisimple}\}/\cong$$
\end{defn}

We give a very crude form of Local Langlands Correspondence: There is a map 
$$\mathrm{LL}_G: \Pi(G) \to \Phi(G)$$ all of whose fibers are finite non-empty subsets ("L-packets") of $\Pi(G)$ and satisfying various requirements without which the statement will not be meaningful. We omit the details here.

\begin{exmpl}
For $G=GL_n$ such a correspondence is a theorem of Harris-Taylor, in which case $\mathrm{LL}_G$ is a bijection, and one way of characterizing the bijection is via $L$-functions and epsilon factors. The case of a general reductive group is still wide open.
\end{exmpl}

\begin{exmpl}[Unramified LLC]
Let $\Phi_{unr}(G)$ be the set of equivalent classes of unramified homomorphisms 
$\gamma: W_F/I_F \to G^{\vee}(\mathbb C)$ such that $r(\mathrm{Frob})$ is semisimple. It is in bijection with the set of semisimple conjugacy classes in $G^{\vee}(\mathbb C)$.

Let $\Pi_{unr}(G)$ be the set of irreducible smooth $G$-representations $\pi$ such that there exists a connected reductive group scheme $\mathbf{G}$ defined over $\mathcal O$ with generic fiber $G$ and $\pi^{\mathbf{G}(\mathcal O)}\neq 0$. (Note that in general there may exist more than one such reductive integral model $\mathbf{G}$).

The unramified LLC is then the map
\[
 \Pi_{unr}(G) \to \Phi_{unr}(G)
\]
induced by the Satake isomorphism. To be precise, let $\pi \in \Pi_{unr}(G)$, with $\mathbf{G}/\mathcal{O}$ the reductive group scheme such that $\pi^{\mathbf{G}(\mathcal{O})}\neq 0$. Then the Satake isomorphism $\mathcal{H}_{\mathbf{G}(\mathcal{O})} \xrightarrow{\sim} \mathcal{H}_{T(\mathcal{O})}^W$ associates to the $\mathcal{H}_{\mathbf{G}(\mathcal{O})}$-module $\pi^{\mathbf{G}(\mathcal{O})}$ a semisimple conjugacy class in $G^\vee(\mathbb{C})$, and thus an element of $\Phi_{unr}(G)$.
\end{exmpl}

\begin{rmk}[To Example 1.10]
Even in the unramified setting, we can have $L$-packets that are not singletons.  For instance, we can have smooth irreducible unramified representation $\tilde{\pi}$ on $\mathrm{GL}_2(\rats_p)$ such that $\tilde{\pi}|_{\mathrm{SL}_2(\rats_p)}$ is not irreducible; in this case the constituents of the restriction form an $L$-packet. For example, the irreducible unramified representation
$$\mathrm{Ind}_B^{\mathrm{GL}_2(\rats_p)}(\varepsilon|\cdot|^{1/2}\otimes|\cdot|^{-1/2}),$$ where $\varepsilon$ is the unramified character 
$$\rats_p/\ints_p \to \{\pm 1\}$$
$$p \mapsto -1$$,
is reducible when restricted to $\mathrm{SL}_2(\mathbb{Q}_p)$.
\end{rmk}

Back to Satake isomorphism
$$\mathcal S: \mathcal H_K \to R(G^{\vee})\otimes_{\ints}\mathbb C$$
LHS has a basis $c_{\lambda}=\mathbf{1}_{K\lambda(\varpi)K}$, $\lambda\in X_*(T)_+$, RHS has a basis $ch(V(\lambda))$, $\lambda\in X_*(T)_+$ where $V(\lambda)$ is the highest weight representation corresponding to $\lambda$.

\textbf{Motivating Question}: What in $\mathcal H_K$ maps to $ch(V(\lambda))$? A full understanding will come from geometric Satake.

Let us do some warm-up calculation before moving to the proof of Satake Isomorphism.

\begin{lem}
For all $G$, let $\lambda, \mu \in X_*(T)$. Write 
$$K\lambda(\varpi)K=\bigsqcup x_iK$$ We may assume by Iwasawa decomposition that $x_i \in B(F)$ and write $x_i=t(x_i)n(x_i) \in T\cdot N$. Let $\rho$ be the half sum of positive roots and $q$ be the cardinality of the residue field. Then
$$S(c_{\lambda})(\mu(\varpi))=q^{-<\mu,\rho>}\cdot
|\{ i\:|\:t(x_i)\equiv \mu(\varpi) \:\:\mathrm{mod} \:\: T(\mathcal O) \}|$$
\end{lem}
\begin{proof}
$$S(c_{\lambda})(\mu(\varpi))=\delta(\mu(\varpi))^{1/2}\int_N \mathbf{1}_{K\lambda(\varpi)K}(\mu(\varpi)n)dn=q^{-<\mu,\rho>}\sum_i \int_N\mathbf{1}_{x_iK}(\mu(\varpi)n)dn$$
Note that $$\mu(\varpi)n\in x_iK=t(x_i)n(x_i)K \iff n\in \mu(\varpi)^{-1}t(x_i)n(x_i)K=n(x_i)'\mu(\varpi)^{-1}t(x_i)K$$
i.e., $\exists$ $k\in K$ such that $k=t(x_i)^{-1}\mu(\varpi)n(x_i)'^{-1}n$, which is equivalent to 
$t(x_i)^{-1}\mu(\varpi) \in T(\mathcal O)$ and $n(x_i)'^{-1}n \in N(\mathcal O)$.
So
$$S(c_{\lambda})(\mu(\varpi))=q^{-<\mu,\rho>}\sum_{\textrm{i such that $t(x_i)\equiv \mu(\varpi) \:\:\mathrm{mod} \:\: T(\mathcal O)$}}\int_{n(x_i)'N(\mathcal O)}dn$$
$$=q^{-<\mu,\rho>}\cdot
|\{ i\:|\:t(x_i)\equiv \mu(\varpi) \:\:\mathrm{mod} \:\: T(\mathcal O) \}|$$

\end{proof}

Now we turn to $G=PGL_2(F)$, then $\lambda \in X_*(T)_+$ are
$$ne_1^*: t \mapsto 
\begin{pmatrix}
t^n & 0\\ 0&1 
\end{pmatrix} \in PGL_2(F)
$$

Let us compute $S(c_0),S(c_{e_1^*}),S(c_{2e_1^*})$.
It is clear that $S(c_0)=1(=1\cdot [0] \in \mathbb C[X_*(T)])$. For $S(c_{e_1^*})$, decompose 
$$Ke_1^*(\varpi)K=K\begin{pmatrix} \varpi & 0\\ 0&1 \end{pmatrix}K$$
$$=\bigsqcup_{x\in \mathcal O/\varpi} \begin{pmatrix} \varpi & x\\ 0&1 \end{pmatrix}K \cup \begin{pmatrix} 1 & 0\\ 0&\varpi \end{pmatrix}K$$ 
i.e.,
$$S(c_{e_1^*})=[e_1^*]\cdot q^{-<e_1^*,\rho>}|\mathcal O/\varpi|+[e_2^*]\cdot q^{-<e_2^*,\rho>}\cdot 1=q^{1/2}(e_1^*+e_2^*)=q^{1/2}ch(V(e_1^*))$$ (Note $W$-invariance!)

Next, for $S(c_{2e_1^*})$, decompose 
$$K\begin{pmatrix} \varpi^2 & 0\\ 0&1 \end{pmatrix}K=\bigsqcup_{x\in \mathcal O/\varpi^2} \begin{pmatrix} \varpi & x\\ 0&\varpi \end{pmatrix}K \cup
\bigsqcup_{x\in (\mathcal O/\varpi)^{\times}} \begin{pmatrix} \varpi^2 & x\\ 0&1 \end{pmatrix}K
\cup 
\begin{pmatrix} 1 & 0\\ 0&\varpi^2 \end{pmatrix}K$$
So $$S(c_{2e_1^*})=[2e_1^*]\cdot q^{-1}\cdot q^2+[0]\cdot (q-1)+[2e_2^*]\cdot q\cdot 1$$
$$=q[2e_1^*]+(q-1)[0]+q[2e_2^*]\:\:\:\:(\textrm{Note $W$-invariance!})$$
$$=q\cdot ch(V(2e_1^*))-ch(V(0))$$ where $V(2e_1^*))$ is the symmetric square representation of $PGL_2(F)$.

In general, we have
$$q^{n/2}ch(V(ne_1^*))=\sum_{m\equiv n(2),m\leq n}S(c_{me_1^*})$$ for $G=PGL_2(F)$. This identity corresponds via geometric Satake to singularities of affine Schubert variety corresponding to $ne_1^*$.



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