These are some notes for the comprehensible lectures, by Professor Benedict Gross at Columbia (Fall 2011), on interesting results in representation theory and number theory discovered using the local Langlands correspondence (including a motivated introduction to the Langlands program).

TopThe Langlands correspondence

The Langlands program relates two major branches of mathematics: number theory and representation theory. The (conjectural) Langlands correspondence is a dictionary between them, which enables people to translate problems in number theory to representation theory and vice versa. In its initial form, the local Langlands correspondence is a bijection between the irreducible representations of a reductive group $G$ over a local field $k$, and the conjugacy classes of continuous homomorphism from the Galois group of $k$ to the dual group of $G$ $\phi: \Gal(k^s/k)\rightarrow \hat G(\mathbb{C})$. These homomorphisms are called Langlands parameters (to be redefined for three more times later). Local class field theory says that $\Gal(k^s/k)^\mathrm{ab}$ is isomorphic to the profinite completion of $k^\times$, hence can be formulated as the case $G=GL_1$ under the framework of Langlands program. From this point of view, the Langlands program can be regarded as a vast nonabelian generalization of class field theory. In the global case, the Langlands conjecture predicts a correspondence between automorphic representations and Langlands parameters of the global field.

Note that $\Gal(k^s/k)$ is a profinite group and we endow the complex Lie group $\hat G(\mathbb{C})$ with discrete topology, so a Langlands parameters $\phi$ factors through a finite quotient $D=\Gal(K/k)$ of $\Gal(k^s/k)$. So a Langlands parameter is given by a finite subgroup of $H=\hat G(\mathbb{C})$ and an isomorphism $\phi: D\rightarrow H$ up to conjugacy by $N_{\hat G(\mathbb{C})}(H)$. Note that for $k$ a local field, $D$ has a ramification filtration by the inertia group and wild inertia groups $D\rhd D_0\rhd D_1\rhd\cdots\{1\}$, so there are certain restriction on the structure of the finite subgroup $H $.

When $K/k$ is unramified, $D=\Gal(K/k)$ is generated by the Frobenius element and thus a Langlands parameter corresponds to a conjugacy class of finite order in $\hat G(\mathbb{C})$. In particular, this conjugacy class is semisimple and is classified by the points of the Steinberg variety $s\in \hat T/W$. Katz completely classified these semisimple conjugacy classes of finite order using the affine diagram. By the very definition of the dual group, $s$ corresponds to a character $\chi_s: T(k)\rightarrow \mathbb{C}^\times$, and hence a representation $\pi_s$ of $G$ given by the parabolic induction $\pi_s=\Ind_B^G\chi_s$. However, $\pi_s$ is not always irreducible. In other words, in this case a Langlands parameter corresponds to a finite set of irreducible representations of $G(k)$, instead of a single irreducible representation. This finite set is called an $L$-packet.

An invariant related to the $L$-packet associated to a Langlands parameter $\phi $ is the component group of the centralizer $A_\phi=\pi_0(C_{\hat G}(\phi))$. Since $C_{\hat G}(\phi)$ is the stabilizer of the $\hat G$ orbit of $\phi$, a representation of $A_\phi$ can be viewed as a local system on the orbit.

Example 1 For $G=GL_n$, this group is always trivial and the $L$-packet consists of a single irreducible representation. For $G=SL_2$, any conjugacy class of finite order in $\hat G(\mathbb{C})=PGL_2(\mathbb{C})$ is of the form $s=
\begin{bmatrix}
  \zeta_n & 0 \\
  0 & 1
\end{bmatrix}$, where $\zeta_n$ is an $n $-th root of unity. When $n\ge3$, $C_{\hat G}(s)=\hat T=SO_2$ but when $n=2$, $C_{\hat G}(s)=N_{\hat G}\hat T=O_2$. So the component group $A_\phi$ is trivial when $n\ge2$ but is $\mathbb{Z}/2 \mathbb{Z}$ when $n=2$. $\mathbb{Z}/2 \mathbb{Z}$ has two irreducible representations and it turns out when $\phi$ has order 2, the corresponding parabolic induction is reducible and has two irreducible components which together form the $L$-packet. In general, $A_\phi$ is an elementary 2-group for all classical groups and can be $S_3$, $S_4$, $S_5$ etc. for exceptional groups.

A Langlands parameter $\phi$ is called discrete if the centralizer $C_{\hat G}(\phi)$ is finite. These parameters corresponds discrete series representations. Note that no unramified parameter is discrete since $\dim C_{\phi}(s)\ge\rank \hat G\ge1$.

When $K/k$ is tamely ramified, $\Gal(K/k)$ is generated by the Frobenius element $F$ and the generator of the tame ramification $\tau$ satisfying $F\tau F^{-1}=\tau^q$. So a tamely ramified Langlands parameter corresponds to a finite group of $\hat G(\mathbb{C})$ generated by two elements $n,s$ of finite order with relation $nsn^{-1}=s^q$. Suppose $s$ is regular in $\hat T$, i.e., $C_{\hat G}(s)=\hat T$. Since $n \in N_{\hat G}(\hat T)$, it maps to an element in the Weyl group $w\in W$ and $C_{\hat G}(\phi)=\hat T^w$. If $w$ is furthermore elliptic, then $C_{\hat G}(\phi)$ is finite and $\phi$ is a discrete parameter.

Example 2 For $G=SL_2$, we have $W=N_{\hat G}(\hat T)/\hat T=O_2/SO_2=\mathbb{Z}/2 \mathbb{Z}$. If we choose $s$ to be of order at least 3 and $w$ to be the nontrivial, then $\hat T^w$ is a group of order 2 consisting the identity and $
\begin{bmatrix}
  1 & 0 \\
  0 & -1
\end{bmatrix}$. Notice that $n^2=1$, $s^q=sn^{-1}=s^{-1}$, we know that $H=\Im \phi$ is a dihedral group of order $2(q+1)$ and the Langlands parameter $\phi$ should correspond to 2 irreducible representation of $SL_2(k)$. For example, when $k=\mathbb{Q}_2$, we have $q=2$ and $H=S_3$. There is a unique tamely ramified extension $\mathbb{Q}_2(\zeta_3,\sqrt[3 ]{2})/\mathbb{Q}_2$ with Galois group $S_3$. The parameter $\phi:\Gal(\mathbb{Q}_2(\zeta_3,\sqrt[3 ]{2})/\mathbb{Q}_2)\rightarrow S_3\subseteq PGL_2(\mathbb{C})$ corresponds 2 irreducible representations of $SL_2(\mathbb{Q}_2)$ given by compact induction from $K_1=SL_2(\mathbb{Z})$ and $K_2=
\begin{bmatrix}
  2 & 0 \\
  0 & 1
\end{bmatrix}K_1
\begin{bmatrix}
  2^{-1} & 0 \\
  0 & 1
\end{bmatrix}=
\begin{bmatrix}
  * & 2* \\
  2^{-1}* & *
\end{bmatrix}
$ of the characters $K_i\rightarrow SL_2(\mathbb{Z}/2 \mathbb{Z})\cong S_3\xrightarrow{\mathrm{sgn}}\{\pm1\}$.

Wildly ramified representations are even more interesting and contains huge arithmetic information (e.g. Bhargava's recent work). (The video is incomplete at this point.)

TopArtin L-functions

A Langlands parameter is a homomorphism $\phi:\Gal(k^s/k)\rightarrow \hat G(\mathbb{C})$. Before studying them, we need to know both sides of this homomorphism. We shall discuss a bit about the number theory side in this section and the structure of reductive groups in the next section.

The ideas of connecting number theory to representation theory started with Artin. Let $K/k$ be a number field extension and $V$ be a representation of $G=\Gal(K/k)$. Artin defined the Artin $L$-function $$L(V,s)=\prod_{\mathfrak{p}}\det(1-F_{\mathfrak{p}}\mathbb{N}\mathfrak{p}^{-s}|V^{I_p})^{-1}.$$

Example 3 When $V$ is the trivial representation of $G$, $$L(V,s)=\prod_{\mathfrak{p}} (1-\mathbb{N}\mathfrak{p}^{-s})^{-1}=\sum_{I}\mathbb{N}I^{-s}$$ is equal to the Dedekind zeta function $\zeta_k(s)$. In particular, when $k=\mathbb{Q}$, the Artin $L$-function $L(V,s)$ recovers the Riemann zeta function $$\zeta(s)=\prod_p(1-p^{-s})^{-1}=\sum_{n\ge1}\frac{1}{n^s}.$$ Also, since all eigenvalues of $F_{\mathfrak{p}}$ are roots of unity, by comparing to $\zeta(s)$, one knows that $L(V,s)$ convergences on the half plane $\Re s>1$.
Example 4 When $K/\mathbb{Q}$ is a quadratic extension of discriminant $d$ and $V$ is the nontrivial 1-dimensional representation of $G=\Gal(K/\mathbb{Q})$, by the quadratic reciprocity we know that 
\begin{align*}
L(V,s)&=\prod_{p \text{ split}}(1-p^{-s})^{-1}\prod_{p\text{ inert}}(1+p^{-s})^{-1}\\&=\prod_p(1-\chi(p)p^{-s})^{-1}=\sum_{n\ge1}\frac{\chi(n)}{n^s}
\end{align*}
is equal to the Dirichlet $L$-function $L(\chi,s)$ associated to the quadratic character $\chi(n)=\legendre{n}{d}$. In general, Artin noticed that the 1-dimensional representations of $G$ correspond to the Dirichlet $L$-functions by Artin reciprocity. Since the analytic properties of Dirichlet $L$-functions are well understood, Artin conjectured these should be also true for higher dimensional representations of $G$. This is the content of the famous Artin conjecture.
Conjecture 1 (Artin)
  1. $L(V,s)$ has meromorphic continuation to the entire plane. Moreover, the continuation is holomorphic if $V^G=0$ (i.e., $V$ contains no trivial representations).
  2. $L(V,s)$ satisfies a functional equation.

The meromorphic part was proved by Brauer using his induction theorem, however, the holomorphic part is still open. Solving the Artin conjecture was one of the motivation of Langlands program: The holomorphic continuation of an Artin $L$-functions will follow from the holomorphic continuation of the associated automorphic $L$-function, which can be established comparatively easily.

By adding the factors at infinity places, Hecke was able to show the functional equation for the Dedekind zeta function $\zeta_k(s)$ in his thesis (and Tate reproved it using Fourier analysis on adeles in his famous thesis). Let $r_1$ and $r_2$ be the number of real and complex places of $k$ and $\Lambda(s)=(\pi^{-s/2}\Gamma(s/2))^{r_1}(2\pi^{-s}\Gamma(s))^{r_2}\zeta_k(s)$, then $$\Lambda(s)=|d|^{1/2-s}\Lambda(1-s).$$ Artin formulated the functional equations of Artin $L$-functions analogously. Define the local factors 
\begin{equation*}
  L_v(V,s)=
  \begin{cases}
    (2\pi^{-s}\Gamma(s))^{\dim V} & v \text{ real}, \\
    (\pi^{-\frac{s}{2}}\Gamma(\frac{s}{2}))^{\dim V^+}(\pi^{-\frac{s+1}{2}}\Gamma(\frac{s+1}{2}))^{\dim V^-} & v\text{ complex},
  \end{cases}
\end{equation*}
where $V^{\pm}$ are the eigenspaces of the complex conjugation. Then $\Lambda(V,s)=L(V,s)\cdot\prod_{v|\infty}L_v(s)$ satisfies the functional equation (proved by Brauer) $$\Lambda(V,s)=w(V)A(V)^{1/2-s}\Lambda(V^*,1-s).$$ Here $w(V)$ is the root number. It is a complex number with absolute value 1 satisfying $w(V)w(V^*)=1$ but is hard to compute in general. The factor $A(V)=|d|^{\dim V}\cdot \mathbb{N}f(V)$, where $d$ is the discriminant of $k$ and $f(V)$ is the Artin conductor of $V$. The Artin conductor $f(V)$ is is defined locally and has a $\mathfrak{p}$-component iff the inertia group at $\mathfrak{p}$ acts on $V$ nontrivially. More precisely, Let $G\rhd G_0\rhd G_1\rhd \cdots \rhd G_m=0$ be the ramification filtration at $\mathfrak{p}$, then $a(V)=\ord_\mathfrak{p}f(V)=\sum_{i\ge0}\dim(V/V^{G_i})\cdot\frac{\#G_i}{\#G_0}$. Artin proved that this number is actually an integer, thus $f(V)$ is an integral ideal of $k$. In fact, using the full power of local class field theory, Artin found a representation $\mathrm{Art}$ of $G_0$ defined over $\mathbb{C}$ such that $a(V)=\dim \Hom_{G_0}(V,\mathrm{Art})$. The Swan conductor $b(V):=\sum_{i\ge1}\dim(V/V^{G_i})\cdot\frac{\#G_i}{\#G_0}$. So $b(V)\ne0$ only when the wild inertia acts nontrivially.

Notice that $L(V_1\oplus V_2,s)=L(V_1,s)\cdot L(V_2,s)$ and $L(\mathrm{Reg}_G,s)=\zeta_K(s)$ for $K/\mathbb{Q}$ is a Galois extension with $G=\Gal(K/\mathbb{Q})$. We know that $\zeta_K(s)=\prod_{V \text{ irr.}}L(V,s)^{\dim v}$ because the regular representation decomposes as $\mathrm{Reg}_G=\bigoplus_{V \text{ irr.}}(\dim V)\cdot V$. Comparing the factors in the functional equation of $\zeta_K(s)$ and $L(V,s)$, we obtain the conductor-discriminat formula $$|d_K|=\prod_{V \text{ irr.}}f(V)^{\dim V}.$$ The Taylor expansion of $\zeta_K(s)$ contains much arithmetic information from the class number formula. The Stark Conjecture predicts the information coming from the Taylor expansion of the Artin $L$-functions, which is still largely open.

More generally, when $L/\mathbb{Q}$ is not necessarily Galois, we take it Galois closure $K/\mathbb{Q}$ with Galois group $\Gal(K/\mathbb{Q})=G$. Suppose $H\subseteq G$ is the subgroup fixing $L$, then $\zeta_L(s)=L(\Ind_H^G \mathbf{1}_H,s)$. Since $\Ind_H^G \mathbf{1}_H=\bigoplus(\dim V^H)\cdot V$, we have $\zeta_L(s)=\prod_{V} L(V,s)^{\dim V^H}$. In particular, when $G\subseteq S_n$ acts 2-transitively, $\Ind_H^G\mathbf{1}_H$ decomposes as two irreducible components: a copy of trivial representation and a copy of an irreducible representation $V_{n-1}$ of dimension $n-1$. Hence $\zeta_L(s)=\zeta(s)L(V_{n-1},s)$ and $|d_L|=f(V_{n-1})$.

Example 5 Consider $k=\mathbb{Q}_2$. Let $f(x)=x^4-2x+2$ (Eisenstein) and $L=\mathbb{Q}_2[x]/f(x)$, then $L/\mathbb{Q}_2$ is totally ramfied and hence $4|e$. Let $K$ be the Galois closure of $L/k$. The discriminant of $f(x)$ is equal to $2^4\cdot 101$. Since $K$ contains $\mathbb{Q}_2(\sqrt{2^4\cdot 101})$ and 101 is not a square in $\mathbb{Q}_2$, we know that $K$ contains the degree 2 unramified extension of $\mathbb{Q}_2$ and $2|f$. It follows that $G$ is a subgroup of $S_4$ of order at least 8, hence is either $S_4$ or $D_8$. Since $S_4$ does not have a cyclic quotient of order 6, we know that $f\ne6$, thus $f=2$. So either $e=12$ ($G_0=A_4$) or $e=4$ ($G_0=(\mathbb{Z}/2 \mathbb{Z})^2$). Since $\Ind_H^G\mathbf{1}_H=\mathbf{1}_G\oplus V_3$, we know that $a(V_3)=4$ by the conductor-discriminant formula. This rules out the case $G_0=(\mathbb{Z}/2 \mathbb{Z})^2$, where $a(V_3)=3+3+\cdots \ge6$. Therefore $G_0=A_4$ $G_1=(\mathbb{Z}/2 \mathbb{Z})^2$. From $a(V_3)=3+3\cdot\frac{1}{3}+\cdots\ge4$, we know that $G_2=0$. Let $V_3'$ be the orthogonal representation of $G=S_4$ with determinant 1, then $a(V_3')=4$ and $b(V_3')=1$ since it only differs from $V_3$ by the sign character. This orthogonal representation gives us a Langlands parameter $\Gal(K/\mathbb{Q}_2)\rightarrow SO_3(\mathbb{C})$ with Swan conductor 1. It corresponds to the representation $\Ind_I\chi$ of $SL(\mathbb{Q}_2)$ induced from the Iwahori subgroup $I\subseteq SL_2(\mathbb{Z}_2)$ of the character $\chi:\begin{bmatrix}
  a & 2b \\
  c & d
\end{bmatrix}\mapsto (-1)^{b+c}$.

TopArtin L-functions for function fields

Let $k=F(X)$ be a function field over a finite field $F$ with $q$ elements and $g$ be the genus of the curve $X$. Let $k\subseteq K$ be a field extension with Galois group $G$. For $V$ a representation of $G$, we can define the Artin $L$-function $L(V, s)$ similarly to the number field case. In the number field case, the exponential factor in the functional equation of $L(V, s)$ is $|d_k|^{\dim V}\mathbb{N}(f(V))$, where $d_k$ is the discriminant and $f(V)$ is the Artin conductor of $V$. For the function field case, the exponential factor is replaced by $q^{(2g-2)\dim V+\deg(a(V))}$, where $a(V)=\sum_{p}a_{p}(V)\cdot p$ is the conductor divisor. The functional equation is similar but has a lot of cancellation since the residue field is always $\mathbb{F}_q$. For example, 
\begin{align*}
\zeta_{F(t)}(s)&=(1-q^{-s})^{-1}\prod_{\mathrm{irr. monic }f}(1-q^{-\deg f\cdot s})^{-1} \\
&=(1-q^{-s})^{-1}\sum_{n\ge0}\#\{\text{monic } f \text{\ of\ } \deg n\}q^{-ns} \\
&=(1-q^{-s})^{-1}\sum_{n\ge0}q^n\cdot q^{-ns} \\
&=\frac{1}{(1-q^{-s})(1-q^{1-s})}
\end{align*}
In general, for a curve of genus $g$, the zeta function will be of the form $$\zeta(s)=\frac{P_{2g}(q^{-s})}{(1-q^{-s})(1-q^{1-s})},$$ which is a rational function of degree $2g-2$ in $q^{-s}$. The general fact that Weil discovered is that the Artin $L$-function $L(V,s)$ is a rational function of $q^{-s}$ of degree $(2g-2)\dim V+\deg(a(V))$ (e.g., $\zeta(s)$ corresponds to the trivial representation and we have $\dim V=1$ and $\deg a(V)=0$.) Even nicer, if the geometric inertia group $J=\Gal(K:F'(X))$ (where $F'$ is the closure of $F$ in $K$) has trivial invariants, namely $V^J=0$, then $L(V,s)$ is actually a polynomial in $q^{-s}$. In particular, when $g(X)=0$ and $V^J=0$, then the degree $-2\dim V+\deg(a(V))\ge0$, thus $\deg (a(V))\ge 2\dim V$ and the equality holds if and only if $L(V,s)=1$. Some interesting things happen even in this seemingly silly case.
Example 6 Let $k=F(t)$ with $\Char(F)\ne2$ (corresponding to the projective line $\mathbb{P}(t)$) and $K=F(\sqrt{t})$ be the quadratic extension of $k$ (corresponding to the curve $y^2=t$). Let $V$ be the non-trivial irreducible representation of $G$. Assume $G=J$, then by the above discussion we know $$L(V,s)=\frac{\zeta_K(s)}{\zeta_k(s)}=1.$$ Then the inertia group is non-trivial for $t=0,\infty$ (tamely ramified as $\Char(F)\ne2$), so $a(V)=(0)+(\infty)$ and $\deg a(V)=2$.
Example 7 In characteristic zero, a finite group acting on $\mathbb{P}^1$ can only be a finite subgroup of $SO(3)\subseteq PGL_2(\mathbb{C})$, which can only be a cyclic group, dihedral group, $A_4$, $S_4$ or $A_5$. But in positive characteristic, the finite automorphism group $PGL_2(q)$ of $\mathbb{P}^1$ over $F=\mathbb{F}_q$ is huge. Let $\mathbb{P}^1\rightarrow \mathbb{P}^1$ be the covering obtained by taking the quotient of $\mathbb{P}^1$ by the group $G=PGL_2(q)$. To find the Artin conductor, let us find the $G$-orbits on $\mathbb{P}^1(\overline{F})$. $\mathbb{P}^1(\mathbb{F}_q)$ is a $G$-orbit with stabilizer the Borel subgroup $B$ of $G$, which has order $q(q-1)$. $\mathbb{P}^1(\mathbb{F}_{q^2})-\mathbb{P}^1(\mathbb{F}_q)$ is a $G$-orbit with stabilizer the non-split torus $T_{q+1}$, which has order $q+1$. All other orbits are free since a fixed point under $G$ necessarily satisfies a quadratic equation over $\mathbb{F}_q$. Choose our third orbit to be a free orbit $\mathbb{P}^1(\mathbb{F}_{q^3})-\mathbb{P}^1(\mathbb{F}_{q})$ (note $\mathbb{P}^1(\mathbb{F}_{q^2})\cap \mathbb{P}^1(\mathbb{F}_{q^3})=\mathbb{P}^1(\mathbb{F}_{q})$).

Let us assume those three orbits are mapping to $\infty, 0, 1$ correspondingly. For the first orbit, there is no residue field extension, so everything is in the inertia group $G_0=B$. It turns out that the ramification filtration is $G_0=B\rhd G_1=U\rhd G_2=1$, where $U$ is the unipotent radical which has order $q$. For the second orbit, the inertia group $G_0=T_{q+1}$ and there is no wild ramification. For the third orbit, $G_0=1$.

By the previous discussion, we know $L(V,s)=1$ for any nontrivial representation of $G$ and $\deg a(V)=2\dim V$. But the Artin conductor $$a(V)=\dim (V/V^T)\cdot(0)+\left((\dim (V/V^B)+\frac{1}{q-1}\dim(V/V^U)\right)\cdot(\infty).$$ Now one can have a lot of fun to verify the equality $\deg a(V)=2\dim V$ for various representations of $G=PGL_2(q)$. We have the principal series representations with $\dim V=q+1$, Steinberg and twisted Steinberg representations with $\dim V=q$, and discrete series representations with $\dim V=q-1$. For the principal series representations, we have $\dim V^T=1$, $\dim V^B=0$, $\dim V^U=2$. We verify that $\deg a(V)=q+q+1+1/(q-1)\cdot (q-1)=2q+2=2\dim V$. All these exotic groups acting on $\mathbb{P}^1$ in positive characteristic is precisely because of wild ramification. There are enormous numbers of examples (e.g., all the Deligne-Lusztig curves) where the covering of $\mathbb{P}^1$ is only ramified at two points and is only wildly ramified at one point.

TopClassification of reductive groups

TopOver algebraically closed fields

Over an algebraically closed field $k=\bar k$ (of any characteristic), it is well known that the reductive groups are classified by their root data. Let $G$ be a reductive group over $k$, then all maximal tori of $G$ are conjugate. Fix a maximal torus $T\subseteq G$. Let $M=\Hom(T,\mathbb{G}_m)$ be the character group. Then the restriction to $T$ of the adjoint representation of $G$ on $\mathrm{Lie}(G)$ decomposes into pieces $\mathrm{Lie}(G)=\mathrm{Lie}(T)\oplus_{\alpha\in \Phi} \mathfrak{g}_\alpha$, where $\Phi=\Phi(G, T)\subseteq M$ is the set of roots and each $\mathfrak{g}_\alpha$ is one-dimensional. Fixing a root $\alpha$, one can find a homomorphism $SL_2\rightarrow G$ (up to conjugacy by $T$) such that $\diag(\lambda,\lambda^{-1})\rightarrow T$ and the Lie algebra of $
\begin{bmatrix}
  1 & * \\
  0 & 1
\end{bmatrix}$ maps into $\mathfrak{g}_\alpha$ and that of $
\begin{bmatrix}
  1 & 0 \\
  * & 1
\end{bmatrix}$ maps into $\mathfrak{g}_{-\alpha}$. So we can associate a coroot $\alpha^\vee\in M^\vee=\Hom(\mathbb{G}_m, T)$ to $\alpha$ and by composing $\alpha^\vee$ with $\alpha$ we get the pairing $\langle \alpha,\alpha^\vee\rangle=2$. Even better, $n_\alpha(x)$, the image of $
\begin{bmatrix}
  0 & x \\
  -1/x & 0
\end{bmatrix}$ lies in the normalizer $N_G(T)$. The quadruple $M_G=(M,\Phi, M^\vee, \Phi^\vee)$ satisfies the reduced root datum axioms:

  1. There is a bijection between $\Phi\rightarrow\Phi^\vee, \alpha\mapsto\alpha^\vee$ and the pairing $\langle \alpha,\alpha^\vee\rangle=2$.
  2. If $\alpha\in \Phi$, then the only integral multiple $n\alpha\in \Phi$ is $n=\pm1$.
  3. The simple reflection $s_\alpha: M\rightarrow M:m\mapsto m-\langle m,\alpha^\vee\rangle \alpha$ carries $\Phi$ into itself and the induced map of $M^\vee$ carries $\Phi^\vee$ into itself.

The element $n_\alpha(x)$ acts on $M$ as the Weyl group $N_G(T)/T$ acts on $M$. Actually it acts via the simple reflection $s_\alpha$ and all the simple reflections permute the roots and generate the Weyl group. The last axiom extremely limits the possibility of root data (e.g.s there are only three rank two root systems) and allows the classification of the root data. The isomorphism theorem and the existence theorem ensures a reductive group is determined by its root datum up to isomorphism and for every root datum there exists a reductive group giving rise to it. The dual group $\hat G$ is defined by switching the role of roots and coroots, which is again only defined up to isomorphism. We can not even talk about conjugacy classes in the dual since it may have outer automorphisms.

Example 8 
\begin{center}
  \begin{tabular}[h]{|c|c|}\hline
    $G$ & $\hat G$\\\hline
    $GL_n$ & $GL_n$\\
    $PGL_n$ & $SL_n$\\
    $SO_{2n}$ & $SO_{2n}$\\
    $SO_{2n+1}$ & $Sp_{2n}$\\\hline
  \end{tabular}
\end{center}

The isomorphism tells us that $\Aut(G,T)\twoheadrightarrow\Aut(M_G)$, however, $\Aut(M_G)$ is usually not equal to $\Aut(G,T)$. For example, when $G=GL_2$, we know that $\Aut(G,T)\cong N_G(T)$ but $\Aut(M_G)\cong \mathbb{Z}/2 \mathbb{Z}$.

When $G$ is a torus of dimension $n $, $\Phi=0$ and $\Aut(G)=\Aut(M)=GL_n(\mathbb{Z})$. On the contrary, when $G$ is semisimple, $\mathbb{Z}\Phi\subseteq M $ and $M \subseteq (\mathbb{Z}\Phi^\vee)^*$ are both of finite index, and the position of $M$ between $\mathbb{Z}\Phi$ and $(\mathbb{Z}\Phi^\vee)^*$ determines the isogeny class of $G$. The Cartier dual of $Z(G)$ is $M/\mathbb{Z}\Phi$. In particular, when $\mathbb{Z}\Phi= M$, then $Z(G)=1$, namely $G$ is adjoint if and only if $M=\mathbb{Z}\Phi$. On the contrary, $G$ is simply-connected if and only if $M=(\mathbb{Z}\Phi^\vee)^*$. More generally, $\mathbb{Z}\Phi\hookrightarrow M$ splits if and only if $Z(G)$ is connected and $\mathbb{Z}\Phi^\vee\hookrightarrow M^\vee$ splits if and only if the derived group of $G$ is simply connected.

Given a root datum $R$, the simple reflections generate the Weyl group $W$, which sits inside the automorphism group $\Aut(R)$. We also have a canonical automorphism $-1\in \Aut(R)$. Sometimes $-1\not\in W$ (e.g. for $GL_n$), but sometimes $-1\in W$ (e.g. for $G_2$). The notion of Borel subgroups come into play in order to understand this problem. Fixing a Borel subgroup $B\supseteq T$ gives more structure on the root datum by looking at the action of $T$ on $\mathrm{Lie}(B)$, namely it picks out the positive roots $\Phi^+(B,T)$ with respect to $B$ and also determines a root basis $\Delta\subseteq \Phi^+$. The $6$-tuple $BR=(M, \Phi, \Delta, M^\vee, \Phi^\vee,\Delta^\vee)$ is called the based root datum and the Weyl group $W$ acts simply transitively on $BR$. So $\Aut(BR)$ and $W$ has trivial intersection inside $\Aut(R)$. In fact, it turns out that $\Aut(R)=W\rtimes\Aut(BR)$.

Notice that the exact sequence $1\rightarrow T\rightarrow N_G(T)\rightarrow W$ does not split ($s_\alpha\in W$ has order two, but $n_\alpha(x)$ may have order 4), hence $W$ does not act on $G$ itself. The amazing thing is that the other part $\Aut(BR)$ actually acts on $G$. A pinning is a choice of $(G, T, B, \{X_\alpha\})$, where $X_\alpha$ is a basis of the simple root space $\mathfrak{g}_\alpha$. Any inner automorphism preserving a pinning must be trivial. A theorem of Chevalley states that $\Aut(G)\cong \Inn(G)\rtimes \Theta$, where $\Theta$ is the group of pinned automorphisms for a fixed pinning, which is in fact equal to $\Aut(BR)$.

TopOver arbitrary fields

We now understand the classification of reductive groups over an algebraically closed field and the automorphism group $\Aut(G)$. Serre gave a method using Galois cohomology and descent to classify reductive groups over an arbitrary field $k$.

For tori, a theorem of Grothendieck ensures that if two tori $T$ and $T_0=\mathbb{G}_m^n$ over $k$ become isomorphic over $\bar k$, then they are actually isomorphic over a finite separable extension of $k$ (i.e., the scheme of tori is smooth). For every isomorphism $f: T_0\rightarrow T$ over $k^s$, conjugating $f$ by $\sigma\in \Gal(k^s/k)$ gives rise to an automorphism $a_\sigma$ of $T_0$ over $k^s$ satisfying $\sigma(f)=f\circ a_\sigma$. This assignment is a 1-cocycle as $a_{\sigma\tau}=a_\sigma\sigma(a_\tau)$ and a different isomorphism $f':T_0\rightarrow T$ over $k^s$ gives rise to $a'$ which is equal to $a$ up to a 1-coboundary. In fact, if $f'=f\circ b$ then $a_\sigma'=b^{-1}a_\sigma\sigma(b)$. Even better, as a torus has no outer automorphism, we know $\Aut(T_0)(k^s)$ and $\Aut(T_0)(k)$ have no difference since both of them are isomorphic to $\Aut(M)=GL_n(\mathbb{Z})$. So the above assignment in fact gives a homomorphism $\Gal(k^s/k)\rightarrow GL_n(\mathbb{Z})$ (up to conjugacy), in other words, an integral Galois representation of rank $n$. Conversely, for any such Galois representation on $M$, we obtain a desired torus $T$ with $k$-points $T(k)=((k^s)^\times\otimes M^\vee)^{\Gal(k^s/k)}$. Hence the category of tori isomorphic to $T_0$ is equivalent to the category of integral $\Gal(k^s/k)$-representations of rank $n $.

Example 9 For $n=1$, we obtain the homomorphism $a:\Gal(k^s/k)\rightarrow \mathbb{Z}^\times\cong\{\pm1\}$. When $a$ is trivial, $T$ is split over $k$. When $a$ is nontrivial, $\ker a$ fixes a separable quadratic extension $E$ with Galois group $\{1,\sigma\}$ and $(E^\times\otimes \mathbb{Z})^{\{1,\sigma\}}\cong\{\alpha\in E^\times: \alpha\cdot\sigma(\alpha)=1\}$. Every such torus can be viewed as a subgroup of $SL_2$ by considering its action on $E=k^2$. For $k=\mathbb{R}$, we get a split torus $\mathbb{R}^\times$ and a nonsplit torus $S^1$. For $k=\mathbb{F}_q$, we obtain a split torus $\mathbb{F}_q^\times$ of order $q-1$ and a nonsplit torus $\mathbb{F}_{q^2,\mathbb{N}=1}$ of order $q+1$.
Example 10 For $n=2$ and $k=\mathbb{R}$, we obtain the conjugacy class of a homomorphism $a: \Gal(\mathbb{C}/\mathbb{R})\rightarrow GL_2(\mathbb{Z})$. Let $A$ be the image of the nontrivial element under $a$. Then we get the tori $(\mathbb{R}^\times)^2$, $\mathbb{C}^\times$, $S^1\times \mathbb{R}$, and $(S^1)^2$ corresponding to $A=
\begin{bmatrix}
  1 & 0 \\
  0 &1
\end{bmatrix}$, $\begin{bmatrix}
  0 & 1 \\
  1 & 0
\end{bmatrix}$, $\begin{bmatrix}
  1  & 0 \\
  0 &-1
\end{bmatrix}$ and $\begin{bmatrix}
  -1 & 0 \\
  0 & -1
\end{bmatrix}$. For $n=2$ and $k=\mathbb{F}_q$, we obtain the homomorphism $a: \hat{\mathbb{Z}}\rightarrow GL_2(\mathbb{Z})$. Its image has order 1,2,3,4,6. For order 2, we obtain the four cases as above. For each of the other orders, we obtain a unique torus.

Now let us consider a general reductive group $G$ over $k$. Given a root datum, Chevalley showed that there exists a unique split group $G_0$ (called the Chevalley group) over $k$ with that root datum (even a group over $\mathbb{Z}$ proved by Grothendieck). Then the same argument as in the case of tori shows that the groups defined over $k$ that are isomorphic to $G_0$ over $k^s$ are classified by the pointed set $H^1(k, \Aut(G_0)(k^s))$. However, due to the existence of $\Inn(G)$ for general $G$, these cohomology classes are no longer homomorphisms as in the case of tori. Anyway, we can map any class $c_G\in H^1(k, \Aut(G_0)(k^s))$ to $d_G\in H^1(k, \Out(G_0)(k^s))$ given by the map $ \Aut(G_0)(k^s)\rightarrow\Out(G_0)(k^s)$ and $d_G$ corresponds to a homomorphism $\Gal(k^s/k)\rightarrow \Out(G_0)(k^s)=\Aut(BR)$. Since $\Aut(BR)$ can be viewed as a subgroup of $\Aut(G)$ , this map $c_G\mapsto d_G$ is actually surjective. Such a form $G_q$ corresponding to $q: \Gal(k^s/k)\rightarrow \Aut(BR)$ is called quasi-split. Moreover, the fiber over $q$ can be identified as the image of $H^1(k, \Inn(G_q))$ (inner forms) in $H^1(k, \Aut(G))$.

Example 11 Let $k=\mathbb{F}_q$ be a finite field. By Lang's theorem, for any connected algebraic group $G$ over $k$, we have $H^1(k, G)=1$. So there is no inner form and $H^1(k, \Aut(G_0))$ is determined by $\Hom(\Gal(k^s/k),\Aut(BR))$ up to conjugacy, in other words, conjugacy classes of finite order in the group $\Aut(BR)$. So the forms over a finite field are classified by pure combinatorical data.
Example 12 Let $k$ be a local field. Kneser proved that $H^1(k, \Inn(G_q))\cong  H^2(k, A_q)$, where $A_q=\ker(G^\mathrm{sc}\rightarrow G^\mathrm{ad}_q=\Inn(G_q))$ is a finite commutative group scheme over $k$ of multiplicative type. By Tate duality, we have a perfect pairing $H^2(k,A_q)\times H^0(k, A_q^\vee)\rightarrow H^2(k, \mathbb{G}_m)\cong \mathbb{Q}/\mathbb{Z}$. Hence $H^1(k, \Inn(G_q))$ is the Pontryagin dual of $(M/\mathbb{Z}\Phi)^{\Gal(k^s/k)}$, where $(M,\Phi)$ is the root system of $G^\mathrm{sc}$. In particular, for $G_q=GL_n$, we obtain $H^1(k, PGL_n)= H^2(k,\mu_n)= H^0(k, \mathbb{Z}/n \mathbb{Z})^\vee=\mathbb{Z}/n \mathbb{Z}$.
Example 13 Let $k$ be a global field. We have the Hasse principle saying that the map $H^1(k, G_q^\mathrm{ad})\rightarrow\bigoplus_v H^1(k_v, G_q^\mathrm{ad})$ into the local Galois cohomologies is injective.

TopLanglands dual groups and Langlands parameters

Let $G$ be a reductive group over an arbitrary field $k$. Then $G$ gives a quasi-split form corresponding to the conjugacy class of a homomorphism $q: \Gal(k^s/k)\rightarrow \Aut(BR)$. Let $E$ be the fixed field of $\ker q$. Then $E/k$ is a finite separable Galois extension of $k$ and splits the maximal torus of $G_q$. $\Aut(BR)$ is isomorphic to the pinned automorphism of $G$, which is also isomorphic to the pinned automorphism of $\hat G$, so we obtain an action of $\Gal(k^s/k)$ on $\hat G$.

Definition 1 The Langlands dual group $^LG$ is defined to be $\hat G\rtimes \Gal(E/k)$, where the connected component $\hat G$ is the complex algebraic group with the root datum dual to $G$ and $\Gal(E/k)$ acts on $\hat G$ as pinned automorphisms (fixing a pinning). Note that $^LG$ only depends on the quasi-split form of $G$.
Definition 2 A Langlands parameter is (initially) a homomorphism $\phi:\Gal(k^s/k)\rightarrow{}^LG$ such that we get the standard surjection $\Gal(k^s/k)\rightarrow \Gal(E/k)$ when composing $\phi$ with the projection $^LG\rightarrow \Gal(E/k)$. Two Langlands parameters are called equivalent if they are conjugate by some $g\in \hat G$.
Example 14 When $G$ is split or an inner form, we have $q=1$, so $^LG=\hat G$. For $G=GL_n$, $^LG=GL_n(\mathbb{C})$ and a Langlands parameter is a $n $-dimensional complex representation of $\Gal(k^s/k)$. For $G=SL_2$, $^LG=PGL_2(\mathbb{C})=SO_3(\mathbb{C})$ and a Langlands parameter is a 3-dimensional orthogonal representation of $\Gal(k^s/k)$ with determinant 1. More generally, for $G=Sp_{2n}$, $^LG=SO_{2n+1}$ and a Langlands parameter is a $2n+1$-dimensional orthogonal representation of $\Gal(k^s/k)$ with determinant 1.
Example 15 For $G=U_n$, $^LG=GL_n(\mathbb{C})\rtimes \Gal(E/k)$. The action of $\Gal(E/k)$ is not by the inverse transpose. Write $GL_n(\mathbb{C})=GL(M)$, where $M$ is an $n $-dimensional vector space. Let $W$ be an orthogonal ($n $ odd) or symplectic ($n$ even) space of dimension $2n$ such that $M$, $M^\vee$ are the maximal isotropic space of dimension $n $ and $W=M\oplus M^\vee$. Then $GL(M)$ is the Levi subgroup of the parabolic stabilizing $M$. When $n $ is odd, the normalizer of $GL(M)$ in $O(W)$ is equal to $^LG$, so a Langlands parameter is an $n $-dimensional orthogonal representation.
Example 16 Let $T$ be a torus of dimension 1. For $T\cong \mathbb{G}_m$, $^LT=GL_1(\mathbb{C})=\mathbb{C}^\times$. For $T\cong U_1(E/k)$, $^LT=\mathbb{C}^\times\rtimes \Gal(E/k)$, where $\Gal(E/k)$ acts on $\mathbb{C}^\times$ by $\sigma z\sigma^{-1}=z^{-1}$. More generally, the Langlands dual of a torus $T$ is given by $^LT=(X^*(T)\otimes \mathbb{C}^\times)\rtimes \Gal(E/k)$, where $X^*(T)$ is the character group. As $X^*(T)\otimes \mathbb{C}^\times$ is abelian, we know that a Langlands parameter $\phi: \Gal(k^s/k)\rightarrow (X^*(T)\otimes \mathbb{C}^\times)\rtimes \Gal(E/k)$ is exactly a 1-cocycle of $\Gal(k^s/k)$ with values in $X^*(T)\otimes \mathbb{C}^\times$ and the equivalence relation is exactly the coboundary condition, hence Langlands parameters correspond to the classes in $H^1(k, X^*(T)\otimes \mathbb{C}^\times)$. From the exact sequence $$1\rightarrow X^*(T)\rightarrow X^*(T)\otimes \mathbb{C}\xrightarrow {e^{2\pi i}} X^*(T)\otimes \mathbb{C}^\times\rightarrow 1,$$ we obtain an isomorphism $H^1(k, X^*(T)\otimes \mathbb{C}^\times)\cong H^2(k, X^*(T))$, as $X^*(T)\otimes \mathbb{C}$ is a complex vector space and all its higher Galois cohomologies are trivial. On the other hand, the Galois pairing $X^*(T)\otimes T(k^s)\rightarrow (k^s)^\times$ induces the cup product pairing $$H^2(k, X^*(T))\otimes H^0(k, T)\rightarrow \mathrm{Br}(k).$$ By local class field theory, $\mathrm{Br}(k)=\mathbb{Q}/\mathbb{Z}$ for $k$ a local $p$-adic field. In this case, Tate proved that the above induces a perfect pairing between $H^2(k, X^*(T))$ and $\widehat {T(k)}$ (the profinite completion of $H^0(k,T)=T(k)$). Therefore a Langlands parameter of $T$ is nothing but a homomorphism $T(k)\rightarrow S^1$ of finite order. In order to get all continuous homomorphisms $T(k)\rightarrow \mathbb{C}^\times$, Langlands replaced $\Gal(k^s/k)$ by the Weil group $W(k)$ so that we can take the Frobenius to anywhere we like.
Definition 3 A Langlands parameter is a homomorphism $\phi: W(k)\rightarrow{}^LG$ such that we get the standard surjection $W(k)\rightarrow \Gal(E/k)$ when composing $\phi$ with the projection $^LG\rightarrow \Gal(E/k)$ .

Under this new and better definition, the Langlands parameters of $T$ are exactly all irreducible complex representation of $T(k)$.

Example 17 There are two more local fields: the real numbers and complex numbers. Their Galois groups are not very interesting, but their Weil groups are. It turns out $W(\mathbb{C})=\mathbb{C}^\times$ and $W(\mathbb{R})=N_{\mathbb{H}^\times}(\mathbb{C}^\times)=\mathbb{C}^\times\cup \mathbb{C}^\times j$, where $\mathbb{H}=\mathbb{R}+\mathbb{R}i+\mathbb{R}j+\mathbb{R}ij$ is the Hamiltonians. In the latter case, we have a (nonsplit) exact sequence $1\rightarrow \mathbb{C}^\times\rightarrow W(\mathbb{R})\rightarrow\Gal(\mathbb{C}/\mathbb{R})\rightarrow1$. Since $W(\mathbb{R})$ has an abelian subgroup of index two, its irreducible representations have dimension either 1 or 2. Since $W(k)^\mathrm{ab}=k^\times$ always holds by local class field theory, the 1-dimensional representations of $W(\mathbb{R})$ are the characters of $\mathbb{R}^\times$ (e.g. the trivial or the sign character). Let $\alpha\in \frac{1}{2}\mathbb{Z}$, we define $\chi_\alpha: \mathbb{C}^\times\rightarrow S^1: z\mapsto (z/\bar z)^\alpha=z^{2\alpha}/|z|^{2\alpha}$ and $W_\alpha=\Ind_{\mathbb{C}^\times}^{W(\mathbb{R})}\chi_\alpha$ be a 2-dimensional representation of $W(\mathbb{R})$. One can check that $W_\alpha\cong W_{-\alpha}$ and is irreducible unless $\alpha=0$ ($W_0=1\oplus \mathrm{sgn}$). The representation $W_\alpha$ is self-dual and is symplectic if $\alpha\in \frac{1}{2}\mathbb{Z}\setminus\mathbb{Z}$ and is orthogonal if $\alpha\in \mathbb{Z}$.
Example 18 Here comes a piece of evidence of the Langlands correspondence. The two real forms $PGL_2(\mathbb{R})=SO_{1,2}(\mathbb{R})$ and $SO_3(\mathbb{R})$ of $PGL_2(\mathbb{C})$ has the same Langlands dual group $^LG=SL_2(\mathbb{C})$. The Langlands parameters $W(\mathbb{R})\rightarrow ^LG=SL_2(\mathbb{C})$ are 2-dimensional symplectic representations and the irreducible ones are exactly $W_\alpha$ parametrized by the half integers $\frac{1}{2},\frac{3}{2},\ldots$ in the previous example. These Langlands parameters correspond to the discrete series representations of $PGL_2(\mathbb{R})$ of weights $2,4,\ldots$ and irreducible representations of $SO(3)$ of dimensions $1,3,\ldots$. More generally, the Langlands parameters $W_{\alpha_1}\oplus \cdots W_{\alpha_n}$, $\alpha_1>\cdots\alpha_n>0$ correspond to the irreducible representations of $SO_{2n+1}(\mathbb{R})$ of highest weight $(\alpha_1,\ldots,\alpha_n)-\rho$, where $\rho=(n-\frac{1}{2},n-\frac{3}{2},\ldots,\frac{1}{2})$.

TopUnramified parameters and unramified representations

Another piece of evidence of the local Langlands correspondence was discovered for a family of so called unramified representations for $p$-adic groups.

Example 19 Consider the multiplicative group $G=\mathbb{G}_m$ over a $p$-adic field $k$. The irreducible representations of $\mathbb{G}_m$ are characters $\chi: k^\times\rightarrow \mathbb{C}^\times$. We say a character $\chi$ is unramified if it is trivial on the units $A^\times$, where $A$ is the ring of integers of $k$. In other words, $\chi$ is uniquely determined by its value on a uniformizer $\pi\in k$. These characters are prevalent as almost all components of a Hecke character $\mathbb{A}^\times/k^\times\rightarrow \mathbb{C}^\times$ are unramified. By local class field theory, $W(k)^\mathrm{ab}\cong k^\times$ and the inertia group $I\subseteq W(k)$ maps to the units $A^\times$, hence an unramified character $\chi$ corresponds to a Langlands parameter $W(k)\rightarrow {}^LG_m$ which is trivial on the inertia group $I $.

Motivated by this:

Definition 4 We say a Langlands parameter $W(k)\rightarrow {}^LG=\hat G\rtimes \Gal(E/k)$ is unramified if its trivial on the inertia group $I $. In particular, $E/k$ is a unramified extension.

Morally speaking, the unramified parameters should correspond to those "unramified" representations of $G(k)$ which are "trivial" on $G(A)$, though $G(A)$ is not normal in $G(k)$ in general.

Example 20 Let $T$ be a torus split by an unramified extension $E/k$. Let $S\subseteq T$ be a maximal split subtorus. Then the cocharacter group $X_*(S)=X_*(T)^{\Gal(E/k)}$. It turns out $T(k)$ has a maximal compact subgroup, which we denote by $T(A)$ (similarly for $S(A)\subseteq S(T)$) and $T(k)/T(A)\cong S(k)/S(A)$. Fixing a choice of uniformizer $\pi\in k$, we have $X_*(S)\cong S(k)/S(A),\phi\mapsto \phi(\pi)$. Hence an unramified representation of $T(k)$ corresponds to an element in $X^*(S)\otimes \mathbb{C}^\times=\hat S(k)$. On the other hand, an unramified parameter is determined by the image of the Frobenius $F$ in $W(k)\rightarrow \hat T$ up to the conjugation of $F$, which can be identified as $\hat T(k)/(1-F)\hat T(k)=\hat S(k)$. So we obtain a bijection between the unramified Langlands parameters and unramified representations of $T(k)$.

We have the following more general notion of unramified representations. Analogously to the 1-dimensional case, almost all components of an automorphic representation are unramified in this sense.

Definition 5 We say $G$ is unramified over $k$ if it is a quasi-split group and split by an unramified extension $E/k$. If $G$ is unramified, then there exists a hyperspecial compact subgroup $G(A)\subseteq G(k)$ (maximal compact subgroup in the sense of volume, not unique even up to conjugation). We say a representation $V$ of $G(k)$ is unramified if $\dim V^{G(A)}=1$. (We will see that any irreducible admissible representation of $G(k)$ satisfies $\dim V^{G(A)}\le1$.)

Let us consider the case when $G$ is split. By Grothendieck's theorem, $G$ is defined over $A$. The hyperspecial group $G(A)$ is simply the $A$-points of $G$. Before proceeding, however, we need to correct our earlier definition of unramified Langlands parameters. In the split case, a Langlands parameter is simply a conjugacy class in the complex group $\hat G$ determined by the image of the Frobenius $F$.

Definition 6 We say a Langlands parameter $W(k)\rightarrow {}^LG=\hat G$ is unramified if it is trivial on the inertia group $I $ and the image of the Frobenius $F$ is semisimple.

The reason comes from the following idea: complex parameters of $W(k)\rightarrow GL_n(\mathbb{C})$ are reflection of families of compatible $\ell$-adic representation $\Gal(k^s/k)\rightarrow GL_n(\mathbb{Q}_\ell)$.

Example 21 Consider the $\ell$-adic Tate module ($\ell\ne p$) of $\mathbb{G}_m$ given by $T_\ell\mathbb{G}_m=\varprojlim_n \mu_{\ell^n}(k^s)\cong \mathbb{Z}_\ell$. Then $T_\ell\mathbb{G}_m$ is a 1-dimensional $\ell$-adic representation of $\Gal(k^s/k)$ and $F$ acts as $q^{-1}$ for each $\ell\ne p$, where $q$ is the cardinality of the residue field. This family corresponds to a complex 1-dimensional representation where $F$ acts as $q^{-1}$.
Example 22 Let $E$ be an elliptic curve over $k$ and consider its $\ell$-adic Tate module $T_\ell E=\varprojlim_n E[\ell^n](k^s)\cong \mathbb{Z}_\ell^2$. Then $T_\ell E$ is a 2-dimensional $\ell$-adic $\Gal(k^s/k)$-representation. When $E$ has good reduction, this representation is unramified and the image of arithmetic Frobenius $F^{-1}$ has an integral characteristic polynomial $x^2-ax+q$ for each $\ell\ne p$, where $a=1+q-\#E(\mathbb{F}_q)$. Weil proved that these images are semisimple, hence are determined by this characteristic polynomial. It is believed that for these $\ell$-adic representations coming from algebraic geometry (e.g. abelian varieties), the Frobenius elements should map to semisimple classes. When $E$ has bad reduction, it turns out that the inertia $I $ acts nontrivially (ramified). Since the wild inertia $I^\mathrm{w}$ is a pro-$p$ group and $GL_n(\ell)$ has a pro-$\ell$ subgroup of finite index, the image of $I^\mathrm{wild}$ in $GL_n(\mathbb{Z}_\ell)$ is finite, hence semisimple. However, the tame inertia $I/I^\mathrm{w}\cong \prod_{\ell\ne p}\mathbb{Z}_\ell$ can map to elements of infinite order. Indeed, Tate showed that for the elliptic curve $y^2=x(x+1)(x-p)$ with multiplicative reduction at $p$, the image of the tame inertia is a subgroup of finite index in $
\begin{bmatrix}
  1 & \mathbb{Z}_\ell \\
  0 & 1
\end{bmatrix}$ for each $\ell\ne p$. Moreover, Grothendieck discovered that there exists a unique nilpotent element $N\in \mathfrak gl_n(\mathbb{C})$ such that the action of the tame inertia is given by the exponential of $N$.

More generally, we will require any Langlands parameter to send $F$ to a semisimple element of $\hat G$ (note that the image of $I $ is of finite order as $I $ is pro-finite, hence is automatically semisimple). To account for the phenomenon of tame inertia we have not seen in the torus case, we introduce an extra factor $SL_2(\mathbb{C})$ in our definition of Langlands parameter as follows.

Definition 7 A Langlands parameter is a homomorphism $\phi: W(k)\times SL_2(\mathbb{C})\rightarrow{}^LG$ which is algebraic on $SL_2(\mathbb{C})$ and sends the Frobenius $F$ to a semisimple element, such that we get the standard surjection $W(k)\rightarrow \Gal(E/k)$ when composing $\phi$ with the projection $^LG\rightarrow \Gal(E/k)$. The group $W(k)\times SL_2(\mathbb{C})$ is sometimes called the Weil-Deligne group of $k$.

Let us return to the split case. If $G$ is split, then the unramified parameters correspond to semisimple conjugacy classes in $\hat G$, hence by conjugation, correspond to elements of the Steinberg variety $\hat T/W$ . The local Langlands correspondence in this case is achieved by the theory of admissible complex representations of the $p$-adic group $G$. Let $K=G(A)$ be the maximal compact subgroup of $G=G(k)$.

Definition 8 An admissible representation of $G$ is a complex representation $V$ of G such that the subgroup fixing any vector $v\in V$ is open (smooth representation) and $V^J$ is finite dimensional for any compact open subgroup $J\subseteq G $. Equivalently, $\Res_K V=\bigoplus_i m_i V_i$, where $V_i$'s are irreducible representations of $K$ (hence are finite dimensional as $K$ is compact). In particular, $V$ is unramified if the trivial representation of $K$ appears in $\Res_KV$.

The Hecke algebra is defined to be $\mathcal{H}_K\otimes \mathbb{C}=C_c^\infty(K\backslash G/K)$, where $\mathcal{H}_K$ is the $\mathbb{Z}$ linear combination of characteristic functions on the double cosets $KgK$. It acts on an admissible representation $V$ via integration and preserves the space $V^K$.

Theorem 1 The map $V\mapsto V^K$ gives a (non-functorial) bijection between all irreducible admissible representations $V$ satisfying $V^K\ne0$ and all simple $\mathcal{H}_K\otimes \mathbb{C}$-modules.
Theorem 2 The Hecke algebra $\mathcal{H}_K$ is commutative. (We say such a pair $(G,K)$ is a Gelfand pairing.)

It follows from this key nontrivial fact that all the simple $\mathcal{H}_K\otimes \mathbb{C}$ have dimension one, hence $\dim V^K\le1$. So to understand the representations of $G$, we need to know more about the structure of the Hecke algebra. From the Cartan decomposition $G=\bigcup_{t\in T(k)/T(A)}KtK$, one looks at the functions $C_c^\infty(T(k)/T(A))^W$ on $T(k)/T(A)\cong X_*(T)$ invariant under the action of the Weyl group.

Theorem 3 (Satake) There is an isomorphism $\mathcal{H}_K\otimes \mathbb{C}\cong \mathbb{C}[X_*(T)]^W$.

An even better version of the Satake isomorphism states that $\mathcal{H}_K\otimes \mathbb{Z}[q^{\pm 1/2}]\cong \mathbb{Z}[q^{\pm 1/2},X_*(T)]^W$.

Example 23 Consider $G=GL_n$. In this case $T(k)/T(A)$ has a basis $\diag(\pi^{i_1},\pi^{i_2},\ldots,\pi^{i_n})$, where $i_1\ge \cdots i_n$ are integers. Hence the Hecke algebra is generated by $T_i=\diag(\underbrace{\pi,\cdots,\pi}_i,1,\cdots,1)$ and $T_n^{-1}$ ($i=1,\ldots n$). In particular, for $n=2$ one can recognize the classical Hecke operators for modular forms.

Since $X_*(T)\cong X^*(\hat T)$, $\mathbb{C}[X_*(T)]^W=\mathbb{C}[X^*(\hat T)]^W$ is the coordinate ring of the Steinberg variety $\hat T/W$ (which can be also identified as the representation ring $R(\hat G)$ of $\hat G$). Thus the characters of $\mathcal{H}_K\otimes \mathbb{C}$ are the points of $\hat T/W$. We now know that the irreducible unramified representations $V$ of $G$ correspond exactly to unramified parameters $W(k)\rightarrow\hat G$. This is the main reason why the dual group $\hat G$ plays such a big role in the whole story.

TopSteinberg representations and supercuspidal representations

Now let us discuss certain classes of ramified local representations.

Let $G$ be any reductive group over a $p$-adic field. Let $(\hat G, \hat B,\hat T,\{\hat X_i\})$ be the pinning we fixed in the definition of the Langlands dual group. The principal regular unipotent element is the element $N=\sum \hat X_i\in\hat{\mathfrak g}$. It determines a homomorphism $SL_2(\mathbb{C})\rightarrow \hat G$ by sending $
\begin{bmatrix}
  0 & 1 \\
  0 & 0
\end{bmatrix}$ to $N$ on the Lie algebra level. Since $N$ is fixed under the action of $\Gal(E/k)$, we obtain a homomorphism $\phi: \Gal(E/k)\times SL_2(\mathbb{C})\rightarrow {}^LG$. This induces the canonical Steinberg parameter $\phi: W(k)\times SL_2(\mathbb{C})\rightarrow{}^LG$. The Steinberg representation of $G$ is the canonical representation associated to this Steinberg parameter.

Example 24 For $G=SL_n$, the principal regular nilpotent element $$N=\begin{bmatrix}
  0 & 1 &   & &   \\
  & 0 & 1 & &  \\
  & & \ddots & & \\
  & & & 0 & 1 \\
  & & & & 0\\
\end{bmatrix}.$$ For $G=SL_2$, $N=
\begin{bmatrix}
  0 & 1 \\
  0 & 0
\end{bmatrix}$ and the parameter coming from an elliptic curve with multiplicative reduction is the Steinberg parameter.

To construct the Steinberg representation, let us begin with the finite field case which Steinberg originally discovered. Let $G$ be a reductive group over $\mathbb{F}_q$, then $G(q)$ is a finite group. Any reductive group over a finite field is quasi-split, hence has a Borel subgroup $B(q)$ and its unipotent radical $U(q)$. It turns out that $U(q)$ is a $p$-Sylow subgroup of $G(q)$. Steinberg discovered a canonical $G(q)$-representation $\mathrm{St}(q)$ of dimensional $\#U(q)$. It restriction to $U(q)$ is the regular representation and has the property that $\mathrm{St}(q)^{B(q)}$ is 1-dimensional. $\mathrm{St}(q)^{B(q)}$ can be viewed as a representation of the Hecke algebra $\mathcal{H}_{B(q)}=\mathbb{C}[B(q)\backslash G(q)/ B(q)]$. We have the Bruhat decomposition $G(q)=\cup_{w\in W} B(q)w B(q)$ and an isomorphism $\mathcal{H}_{B(q)}\cong \mathbb{C}[W]$.

The latter isomorphism can be seen as a $q$-deformation: the group algebra $\mathbb{C}[W]$ is generated by $s_\alpha$ satisfying $(s_\alpha-1)(s_\alpha+1)=0$ (and other relations) where the $\alpha$'s are the simple roots. Analogously, the Hecke algebra $\mathcal{H}_{B(q)}$ is generated by $T_\alpha$ satisfying $(T_\alpha-q)(T_\alpha+1)=0$. $\mathbb{C}[W]$ has two distinguished 1-dimensional representations: the trivial representation $s_\alpha\mapsto1$ and the sign representation $s_\alpha\mapsto-1$. Analogously, $\mathcal{H}_{B(q)}$ also has two distinguished 1-dimensional representations: the trivial representation $T_\alpha\mapsto q$ and the Steinberg representation $T_\alpha\mapsto -1$.

Steinberg's idea extends to the $p$-adic settings. We shall assume $G$ is simply-connected for simplicity. We define the Iwahori subgroup $I\subseteq G(A)$ be the lifting of $B(q)\subseteq G(q)$ and the pro-$p$ unipotent radical $U\lhd I$ be the lifting of $U(q)\lhd B(q)$. These groups play a similar role as the finite field case: the Steinberg representation $\mathrm{St}$ has the property $\dim\mathrm{St}^I=1$. However, the Hecke algebra $\mathcal{H}_I=C_c^\times(I\backslash G(k)/I)$ is noncommutative in this case. Nevertheless, we have the Iwahori decomposition $G(k)=\cup_{w\in W_\mathrm{aff}} IwI$, where the affine Weyl group $W_\mathrm{aff}=N_G(T)(k)/T(A)$ (generated by the simple roots and the lowest root), and an isomorphism $\mathcal{H}_I=\mathbb{C}[W_\mathrm{aff}]$ as a $q$-deformation. Again the Steinberg representation $\mathrm{St}$ corresponds to $T_\alpha\mapsto-1$, which in turn implies that for any large group $P\supsetneq I$ (called a parahoric subgroup), $\mathrm{St}^P=0$.

Since $\mathrm{St}(q)$ appears in $\Res_{G(A)}(\mathrm{St})$ with multiplicity 1, by Frobenius reciprocity, $\Ind_{G(A)}\mathrm{St}(q)$ contains $\mathrm{St}$. However, $\Ind_{G(A)}\mathrm{St}(q)$ also contains other representations containing $\mathrm{St}(q)$ (e.g. many unramified representations) and has infinite length. On the other hand, there exist irreducible representations $V(q)$ such that $\Ind_{G(A)}V(q)=\pi$ is an irreducible representation (called depth-zero supercuspidal) and $\Res_{G(A)}\pi=V(q)$.

Example 25 For $G=SL_2(q)$ we have the trivial representation of dimension 1, the Steinberg representation of dimension $q$, induced representations $\Ind_{B}\chi$ ($\chi$ a character of the split torus) of dimension $q+1$ and discrete series (or supercuspidal) representations of dimension $q-1$. The discrete series are associated to the characters of the non-split torus and are much harder to construct than others. Drinfeld found a way to produce the discrete series by taking the first $\ell$-adic cohomology of the curve $z^{q+1}=x^qy-xy^q$.

Drinfeld's idea was generalized by Deligne-Lusztig for any group. Those supercuspidal representations they constructed are indexed by the characters of anisotropic tori. Anisotropic tori are parametrized by the elliptic conjugacy classes in the Weyl group $W$ (no invariance on the reflection representation). DeBacker and Reeder constructed the Langlands parameters of these Deligne-Lusztig representations as tamely ramified parameters $\phi: W(k)/I^\mathrm{w}\rightarrow \hat G(\mathbb{C})$, where the tame inertia maps a cyclic subgroup of $\hat T$ of order prime to $p$ such that $Z(C)=\hat T$ (regular) and the Frobenius maps to an elliptic class in $W$.

Even the Steinberg parameters and depth-zero parameters are not enough for the global application. We need to construct wild representations, i.e., representations which have even no vectors fixed by $G(A)_1$. We end this section by constructing a class of wild representations called simple supercuspidal representations.

The Frattini subgroup $U^+=[U,U]^p$ is the smallest normal subgroup such that the Frattini quotient group $U/U^+$ is an elementary $p$-group. Moreover, the generators of $U/U^+$ lifts to a generating set of $U$. When $p$ is large enough, Reeder and the speaker can show that $U/U^+\cong \bigoplus\mathbb{F}_q(\alpha_i)$ as a representation of $T(q) $, where $\alpha_i$'s are the affine simple roots generating the affine Weyl group. Motivated by Gelfand-Graev's theory of generic representations over finite fields, we say a character $\chi$ of $U$ is affine generic if it is non-trivial on each line $ \mathbb{F}_q(\alpha_i)$ (and trivial on $U^+$). It turns out $\Ind_U\chi$ has finite length and has a unique component $\pi_\chi$ with trivial central character. This representation $\pi_\chi$ thus constructed is wild.

Example 26 For $G(A)=GL_2(\mathbb{Z}_p)$, we have $$U=
\begin{bmatrix}
  1+p* & * \\
  p* & 1+p*
\end{bmatrix},\quad U^+=
\begin{bmatrix}
  1+p* & p* \\
  p^2* & 1+p*
\end{bmatrix}, \quad G(A)_1=
\begin{bmatrix}
  1+p* & p* \\
  p* & 1+p*
\end{bmatrix}
.$$ So $U/U^+=\mathbb{F}_p\oplus\mathbb{F}_p$ and $\pi_\chi^{G(A)_1}=0$ for $\chi$ a generic character.

TopMotives of reductive groups

Steinberg found a beautiful formula for the order of a reductive group over finite field. As examples, $$\frac{\#GL_n(q)}{q^{n^2}}=(1-q^{-1})(1-q^{-2})\cdots (1-q^{-n}),$$ $$\frac{\#E_8(q)}{q^{\dim E_8}}=(1-q^{-2})(1-q^{-8})(1-q^{-12})\cdots (1-q^{-30}),$$ where the powers can be recognized as the degree of the basic invariant polynomials of the Weyl group. We will generalize Steinberg's formula using motives.

The motive of a reductive group $G$ over $k$ is a collection of Galois representations constructed as follows. Take a quasi-split inner form $G_q$ of $G$ and fix $T\subseteq B\subseteq G_q$ a torus and Borel subgroup of $G_q$ defined over $k$. Then $Y=X^*(T)\otimes \mathbb{Q}$ ($\ell=\dim Y$) is a representation of $\Gal(k^s/k)$ (trivial when $T$ is split) and also a representation of the Weyl group $W=N_{G(k^s)}T(k^s)/T(k^s)$, hence a representation of the semi-direct product $W\rtimes\Gal(k^s/k)$. Let $S(Y)^W$ be $W$-invariants of the symmetric algebra of $Y$, so $\Gal(k^s/k)$ acts on $S(Y)^W$. A theorem of Chevalley showed that $S(Y)^W$ is a polynomial algebra generated by $\ell$ homogeneous algebraically independent polynomials. Let $R_+\subseteq S(Y)^W$ be the ideal consisting of elements of constant term 0, then $R_+/R_+^2\cong\bigoplus_{d\ge1} V_d$ consists of the basic invariant polynomials of $W$ and the degrees $d$ of the basic invariants are uniquely determined. Each $V_d$ is thus a $\Gal(k^s/k)$-representation. Now Steinberg's formula can be rewritten as $$\frac{\# G(q)}{q^{\dim G}}=\prod_{d\ge1} \det(1-Fq^{-d}| V_d).$$

Example 27 For $G=GL_n$, $R_+/R_+^2\cong \bigoplus_{d=1}^nV_d$, where $V_d$ is a trivial Galois representation of dimension 1. Steinberg's formula for $GL_n(q)$ coincides with the easy direct computation. For $G=U_n(E/k)$, $R_+/R_+^2\cong \bigoplus_{d=1}^nV_d$, where $V_d$ is a $\Gal(E/k)$-representation of dimension 1, trivial when $n $ is even and nontrivial when $n $ is odd.

Using the Tate twist $V(n)$ (as a family of $\ell$-adic representation), Steinberg's formula can be also written as $\det(1- F|\bigoplus_{d\ge1}V_d(d))$. More generally:

Definition 9 Let $G$ be a reductive group over an arbitrary field $k$. The motive of $G$ is defined to be $M=\bigoplus_{d\ge1}V_d(1-d)$. (Using $1-d$ instead of $d$ is sometimes more convenient.)

So for $p$-adic fields and global fields, one has corresponding $L$-functions $L(M,s)$ for the motive $M$. Using $V_1=X^*(Z_G^0)$, one can check that $L(M)=L(M,0)$ is finite and nonzero as long as $Z(G)^0$ is anisotropic.

Theorem 4 (Serre) Suppose $k$ is a local field, then there exists a unique invariant measure (i.e., a scalar multiple of the Haar measure) $d\nu$ on $G(k)$ such that for any discrete cocompact torsion free subgroup $\Gamma$ of $G(k)$, $$\int_{\Gamma\backslash G}d\nu=\chi(H^*(\Gamma,\mathbb{Q})).$$ The measure $d\nu$ is called the Euler-Poincare measure.

The problem occurring is the possibility of $d\nu=0$ (e.g. $\chi(\mathbb{Z})=\chi(S^1)=0$). Nevertheless, Serre showed that $d\nu\ne0$ if and only if $G$ contains an anisotropic maximal torus over $k$ (equivalently, $G(k)$ has a discrete series in $L^2(G)$). Moreover, when $k$ is $p$-adic, $d\nu\ne0$ is also equivalent to $Z(G)^0$ being anisotropic, which miraculously matches the condition that $L(M)$ is finite and nonzero.

When $k$ is $p$-adic and $G$ is simply connected, Bruhat-Tits theory gives $\ell+1$ maximal compact subgroup $P_0,\ldots, P_\ell$ containing the Iwahori $I\subseteq G(k)$. Serre found a formula for the Euler-Poincare measure $$d\nu=\sum_{\varnothing\ne S\subseteq \{0,\ldots\ell\}}(-1)^{\# S-1}\frac{dg}{\int_{P_S}dg},$$ where $P_S=\bigcap_{j\in S} P_j$.

Example 28 For $G=SL_2$, $$P_0=SL_2(A)=
\begin{bmatrix}
  * & * \\
  * & *
\end{bmatrix}, \quad P_1=
\begin{bmatrix}
  * & \pi^{-1}* \\
  \pi* & *
\end{bmatrix}, \quad I=
\begin{bmatrix}
  * & * \\
  \pi* & *
\end{bmatrix}.$$ Since $I$ is a subgroup of $P_0$ and $P_1$ of index $q+1$, we know that $$\int_{P_0}dg_A=\int_{P_1}dg_A=1,\quad \int_Idg_A=\frac{1}{q+1},$$ where $dg_A$ is the unique invariant measure assigning the hyperspecial subgroup $SL_2(A)$ measure 1. So Serre's formula shows that $dg_A=(1-q)^{-1}d\nu=L(M)d\nu$. More generally, if $G$ is unramified and simply-connected, we have $dg_A=L(M)d\nu$.

Now consider the global case. Suppose $G$ is a simply-connected reductive group over $\mathbb{Q}$ such that $G(\mathbb{R})$ is compact (so the Euler-Poincare measure assigns $G(\mathbb{R})$ measure 1) and $G(\mathbb{Q}_p)$ is split for every $p$. Let $d\nu$ be the invariant measure such that $\int_{G(\mathbb{Q})\backslash G(\mathbb{A})}d\nu=1$ (called the Tamagawa measure). Let $dg_K$ be the invariant measure assigning $K=G(\mathbb{R})\times \prod_p G(\mathbb{Z}_p)$ measure 1. A great result analogous to the local case is the comparison between the discrete measure $d\nu$ and the compact measure $dg_K$ $$dg_K=\frac{1}{2^{\dim M}}L(M)\cdot d\nu,$$ where $L(M,s)$ is the Artin $L$-function of the motive.

The mass formula then follows from this comparison: $$\sum_{G(\mathbb{Q})\backslash G(\mathbb{A})/K}\frac{1}{\#\Gamma_i}=\frac{1}{2^{\dim M}}L(M),$$ where $\Gamma_i$'s are the finite stabilizers of $G(\mathbb{Q})\backslash G(\mathbb{A})$ under the right $K$-action.

Example 29 For $G=G_2$ split, we have $M=\mathbb{Q}(-1)\oplus \mathbb{Q}(-5)$, therefore $\sum\frac{1}{\#\Gamma_i}=\frac{1}{4}\zeta(-1)\zeta(-5)=\frac{1}{2^6\cdot3^3\cdot7}$. But one orbit has stabilizer $\Gamma=G_2(2)$ of exact order $2^6\cdot3^3\cdot7$, so there is only one orbit and $G(\mathbb{A})=G(\mathbb{Q})\cdot K$. (There are no other way yet to prove this!)

More generally, suppose $G$ is unramified outside a finite set $S$. Let $$dg_S=d\nu_\infty\times\prod_{p\in S}d\nu_p\times\prod_{p\not\in S}dg_A,$$ then $$dg_S=\frac{1}{2^{\dim M}}L_S(M)\cdot d\nu.$$

TopThe trace formula and automorphic representations with prescribed local behavior

For simplicity, we shall assume $G$ is a simply-connected simple group over $\mathbb{Q}$ such that $G(\mathbb{R})$ is compact (to avoid analytical difficulty of the trace formula). Then $G$ is unramified outside a finite set $S$ and for $p\ne S$ we have the hyperspecial subgroup $G(\mathbb{Z}_p)\subseteq G(\mathbb{Q}_p)$. Under this assumption, $G(\mathbb{Q})\subseteq G(\mathbb{A})$ is discrete and cocompact. So $L^2(G(\mathbb{Q})\backslash G(\mathbb{A}))$ decomposes as a Hilbert sum $\bigoplus m(\pi)\pi$ of irreducible unitary representations of $G(\mathbb{A})$ with finite multiplicity.

Let $\phi\, dg$ be a continuous function on $G(\mathbb{A})$ with compact support, it acts on $L^2(G(\mathbb{Q})\backslash G(\mathbb{A}))$ via integration $v\mapsto\int_{G(\mathbb{A})}\phi(g) gv\ dg$. Such a function is a product of local functions $\phi=\prod\phi_v$, where $\phi_v=\mathbf{1}_{G(A_v)} dg_{A_v}$ for almost all places $v $. Each irreducible representation $\pi$ of $G(\mathbb{A})$ can be decomposed into local factors $\pi=\bigotimes_v \pi_v$, where $\pi_v$ is an irreducible unitary representation of $G(\mathbb{Q}_v)$. Since $\mathbf{1}_{G(A_v)} dg_{A_v}$ acts on $\pi_v$ as the projection onto the $G(A_v)$-fixed space, its trace on $\pi_v$ is either 1 (when $\pi_v$ is unramified) or 0 (otherwise). So $\tr(\phi|\pi)=\prod_v\tr(\phi_v|\pi_v)$ is a finite product and the sum $\tr(\phi|L^2(G(\mathbb{Q})\backslash G(\mathbb{A})))=\sum m(\pi)\tr(\phi|\pi)$ is a finite sum.

Next we shall choose a suitable function $\phi$ to pick out specific $\pi$ with prescribed local components. Let $S$ be a finite set of primes. Kottwitz considers the test function $\phi_S$ consisting of $1\cdot dg_\infty$ at $\infty$, $\mathbf{1}_{G(\mathbb{Z}_p)}dg_{\mathbb{Z}_p}$ at $p\not\in S$ and the Euler-Poincare function $\sum_{J\subseteq \{0,\ldots,\ell_p\}}(-1)^{\#J-1}dg_J$ at $p\in S$. Then $\tr(\phi_S|\pi)$ is nonzero if and only $\pi_\infty$ is trivial, $\pi_p$ is unramified for $p\not\in S$ and $\pi_p$ is trivial or Steinberg for $p\in S$ (when $\pi_p$ is unitary, a theorem of Casselman says that $\tr(\sum_{J\subseteq \{0,\ldots,\ell_p\}}(-1)^{\#J-1}dg_J|\pi_p)=\chi(H^*(\pi_p))=0$ unless $\pi_p$ is trivial or Steinberg). When $G(\mathbb{Q}_p)$ is compact, the Steinberg is the trivial representation. By strong approximation if $G(\mathbb{Q}_p)$ is not compact and $\pi_p$ is trivial, then $\pi$ is trivial itself. Consequently, $$\tr(\phi_S|L^2(G(\mathbb{Q})\backslash G(\mathbb{A})))=1+\prod_{p\in S}(-1)^{\ell_p}\sum m(\pi),$$ where the sum runs over all $\pi$ such that $\pi_\infty$ is trivial, $\pi_p$ is Steinberg for $p\in S$ and $\pi_p$ is unramified for $p\not\in S$.

The trace formula computes the trace in terms of orbital integrals: $$\tr(\phi| L^2(G(\mathbb{Q})\backslash G(\mathbb{A})))=\sum_{\gamma\in G(\mathbb{Q})/\sim}\int_{G_\gamma(\mathbb{Q})\backslash G(\mathbb{A})}\phi(g^{-1}\gamma g)dg.$$ So for $\phi=\phi_S$, the trivial conjugacy class of $G(\mathbb{Q})$ contributes to the right-hand-side by $$\int_{G(\mathbb{Q})\backslash G(\mathbb{A})}\phi_S(1)dg=\int_{G(\mathbb{Q})\backslash G(\mathbb{A})}\prod_{p\in S}d\nu_p\cdot\prod_{p\not\in S}dg_{\mathbb{Z}_p}=\frac{1}{2^{\dim M}}L_S(M).$$

Since our test function $\phi_S$ is supported in an open compact subgroup of $G(\mathbb{A})$, if the orbital integral of $\gamma$ is nonzero, $\gamma$ has to be of finite order. However, these integrals of torsion classes $\gamma$ are quite complicated. (There can be infinitely many of them over $\mathbb{Q}$!) People sometimes replace conjugacy classes by stable conjugacy classes in the trace formula to avoid this complexity.

Here we use a trick to simply the trace formula by introducing simple supercuspidal representations at another finite set $T$ of primes disjoint from $S$. For $p\in T$, we choose an affine generic character $\chi_p$ of the pro-$p$ unipotent radical $U_P$ and pick our test function at $p$ to be $\phi_p=\overline{\chi}_pdg_{U_p}$ on $U_p$. When $Z(\mathbb{Q}_p)=1$, $\Ind_{U_p}^{G(\mathbb{Q}_p)}\chi$ is an irreducible simple supercuspidal representation. So by Frobenius reciprocity, we know that the operator $\phi_p$ picks out this simple supercuspidal representation when taking the trace. Suppose $T$ consists of at least two distinct primes $p$ and $q$, if the orbital integral of a conjugacy class $\gamma$ is nonzero, then $\gamma$ is conjugate to elements of $U_p$ and $U_q$, thus must have order $p^n$ and $q^m$ simultaneously, hence $\gamma$ must be trivial. Consequently, the orbital side only consists one term $$\int_{G(\mathbb{Q})\backslash G(\mathbb{A})}\phi_{S,T}(1)dg=\frac{1}{2^P\dim M}L_{S,T}(M).$$ In sum, the Artin $L$-function of the motive counts the automorphic representations of prescribed local components: $\pi_\infty$ is trivial, $\pi_p$ is Steinberg for $p\in S$, $\pi_p$ is the simple supercuspidal associated to $\chi$ for $p \in T$ and unramified elsewhere: $$\frac{1}{2^{\dim M}}L_{S,T}(M)=\prod_{p\in S}(-1)^{\ell_p}\cdot\sum m(\pi).$$

More generally, when $G$ is simple over $\mathbb{Q}$ (but not necessarily simply-connected and $Z(\mathbb{Q}_p)$ is not necessarily trivial), we have the following formula: $$\pm\sum m(\pi)=\frac{1}{2^{\dim M}}L_{S,T}(M)\frac{\# Z(\mathbb{Q})}{\prod_{p\in T}\#Z(p)}\cdot\frac{\# Z(\hat G)^\Gamma}{\prod_{p\in S}\# Z(\hat G)^{\Gamma_p}},$$ where $\Gamma=\Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ and $\Gamma_p\subseteq \Gamma$ is the decomposition group of $p$. Note that if $G$ is simply-connected, then $\hat G$ has trivial center and we recover the previous formula.

Example 30 For $G=E_8$, the above formula gives the number of automorphic representations with prescribed local components:
\begin{equation*}
\begin{split}
\frac{1}{2^8}\zeta(-1)\zeta(-7)\cdots\zeta(-23)\zeta(-29)\prod_{p\in S}(1-p)(1-p^7)\cdots(1-p^{29})\\\cdot\prod_{p\in T}(1-p^2)(1-p^8)\cdots(1-p^{30}).
\end{split}
\end{equation*}
This is a huge number even if the primes in $S$ and $T$ are chosen to be small!
Example 31 For $G=PGL_2$. Let $N$ be the conductor of an irreducible representation $\pi$. Then $\pi_p$ is Steinberg or unramified twisted Steinberg if and only if $p||N$ and $\pi_q$ is simple supercuspidal if and only if $q^3||N$. Then the number of such representations with prescribed local components is $$\frac{1}{24}\prod_{p\in S}(p-1)\prod_{q\in T}(q^2-1)(q-1).$$ These representations correspond to new forms of weight 2 and level $N=\prod_{p\in S}p\prod_{q\in T}q^3$. So we obtain an exact formula for the dimension of $S^{\mathrm{new}}_2(\Gamma_0(N))$.
Example 32 Let us end these lectures by giving an example over a function field. Take $k=\mathbb{F}_q(t)$ of genus 0 and $S=\{0\}$, $T=\{\infty\}$. The trace formula works out similarly, but in the function field case $\zeta(s)=\frac{1}{(1-q^{-s})(1-q^{1-s})}$ and $\zeta_{S,T}(s)=1$, so $L_{S,T}(M)=1$ identically. In other words, there is a unique automorphic representation with such prescribed local behavior. A natural question is to construct a global Langlands parameter $\phi: \Gal(k^s/k)\rightarrow\hat G(\mathbb{Q}_\ell(\mu_p))$ of this unique representation. This parameter is unramified outside $\{\infty,0\}$, hence factors through $\pi_1(\mathbb{G}_m)$. It is tamely ramified at 0 with monodromy a regular unipotent element and wildly ramified at $\infty$. For some groups like $GL_n$, Deligne wrote down these parameters using Kloosterman sums. Frenkel and the speaker constructed complex analogue of these parameters. Later, Heinboth-Ngo-Yun used methods from geometric Langlands theory to construct these parameters in general.