These are my live-TeXed notes for the course Math 270x: Topics in Automorphic Forms taught by Jack Thorne at Harvard, Fall 2013.

Any mistakes are the fault of the notetaker. Let me know if you notice any mistakes or have any comments!

Recommended references for this course:

  • Diamond, Shurman, Introduction to modular forms;
  • Borel, Casselman, Automorphic forms, representations and L-function (Corvallis);
  • Bushnell, Henniart, Local Landlands conjecture for GL(2);
  • Bernstein, Gelbart, Introduction to the Langlands program.


TopModular forms and number fields

The absolute Galois group $\Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ contains huge arithmetic information. One can ask algebraic number theoretic questions like: for which prime $p$, $GL_2(\mathbb{F}_{p^3})$ is a quotient of $\Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ (i.e., there exists a number field with such a Galois group)? Or how to describe Galois extensions of $\mathbb{Q}$ with prescribed local ramification behavior at a prime $p$? Miraculously that these kinds of questions can be answered using modular/automorphic forms and furthermore automorphic forms can be understood by the Langlands dual group ${}^LG$.

Example 1 (Eisenstein series) For $k\ge4$, we define the weight $k$ Eisenstein series $$G_k(\tau)=\sum_{m,n\in  \mathbb{Z},(m,n)\ne0}(m\tau+n)^{-k}.$$ It is easy to show that this series converges absolutely and converges uniformly in any compact subset of $\mathcal{H}$. Notice that $SL_2(\mathbb{Z})$ acts on $\mathbb{Z}^2$ on the right. The stabilizer of $(0,1)$ is $\Gamma_\infty=\left\{
  1 & n \\
  0 & 1
\end{bmatrix}\right\}$. Thus $$G_k(\tau)=\sum_{\Gamma_\infty\backslash \Gamma(1)}j(\gamma,\tau)^{-k}.$$ The weak modularity of $G_k(\tau)$ follows from the modular cocycle property of $j(\gamma,\tau)$. $G_k(\tau)$ is uniformly convergent in $[-1/2,1/2]\times i[1,\infty)$ and thus $$\lim_{\tau\rightarrow\infty}G_k(\tau)=\sum_{n\in \mathbb{Z} ,n\ne0} n^{-k}=2\zeta(k).$$ So $G_k(\tau)$ is bounded as $\gamma\rightarrow \infty$. We have verified that $G_k(\tau)$ is a modular form of weight on $\Gamma(1)$. A standard fact is that $G_k(\tau)$ has a nice $q$-expansion: $$E_k(\tau):=G_k(\tau)/2\zeta(k)=1-\frac{2k}{B_k}\sum_{n\ge1}\sigma_{k-1}(n)q^n,$$ where $B_k$ is the $k$-th Bernoulli number (a rational number).
Example 2 The Ramanujan modular function $\Delta(\tau)$ is defined to be $$\Delta(\tau)=(E_4^3-E_6^2)/1728=q-24q^2+252q^3+\cdots.$$ It is a cusp form of weight 12 on $\Gamma(1)$. $\Delta(\tau)$ is the unique (up to scalar) cusp form on $\Gamma(1)$ of minuscule weight (i.e., there is no cusp forms of weight $\le11$ on $\Gamma(1)$). Moreover, all the coefficients of $q$-expansion of $\Delta$ are indeed integers.

Hecke operators The quotient $\Gamma(1)\backslash(\Gamma(1)
  1 & 0 \\
  0 & p
\end{bmatrix}\Gamma(1))$ has $p+1$ representatives $
  1 & a \\
  0 & p
\end{bmatrix}$, where $a=0,\ldots p-1$ and $
  p & 0 \\
  0 & 1
\end{bmatrix}$. One can then explicitly compute $$T_p(f)=\sum_{n\ge0}a_{np}q^n+p^{k-1}a_n q^{np}.$$ In other words, $$a_n(T_p(f))= a_{np}(f)+p^{k-1}a_{n/p}(f).$$ It follows directly that $T_pT_q=T_qT_p$ for $p\ne q$.

Peterson inner product Fix an integer $k\ge1$. For $f,g$ two holomorphic functions on $\mathcal{H}$, we define $\omega(f,g)=\omega_k(f,g)=f(\tau)\overline{g(\tau)} y^{k-1}dxdy$. One computes that $$\omega(f|_k[\gamma],g)=\gamma^*\omega(f,g|_k[\gamma']), \quad \gamma'{}=\det\gamma\cdot\gamma^{-1}.$$ In particular, $\omega(f,g)$ is $\Gamma(1)$-invariant when $f,g\in M_k(\Gamma(1))$. For $f,g\in S_k(\Gamma(1))$, $\langle f,g\rangle=\int_{\Gamma(1)\backslash\mathcal{H}}\omega(f,g)$ converges and defines a Hermitian positive definite inner product, called the Peterson inner product.

An important fact is that $T_p$ is self-adjoint with respect to the Peterson inner product. Therefore we can simultaneously diagonalize these commuting Hecke operators $\{T_p\}$ to obtain a basis $\{f_i\}$ of $S_k(\Gamma(1))$ such that every $f_i$ is a $T_p$-eigenvector with real eigenvalue for any $p$.

The upshot is that starting from a weight $k$, we obtain real numbers (the $T_p$-eigenvalues) and modular forms $\{f_i\}$ indexed by the prime numbers $p$. We can associate to this basis $\{f_i\}$ Galois representations (or motives) and doing becomes useful for constructing extensions of $\mathbb{Q}$ with prescribed behavior.

Example 3 Let $G=\Gal(\mathbb{Q}(\zeta_p)/\mathbb{Q})$ and $C_k$ be the $k$-th isotypic part of $\Cl(\mathbb{Q}(\zeta_p))\otimes \mathbb{F}_p$. Herbrand and Ribet proved that for $2\le k\le p-3$, it is true that $C_{1-k}\ne0$ if and only if $p\mid B_k$. In particular, when $p\mid B_k$, by class field theory, there exists a unramified $p$-extension of $\mathbb{Q}(\zeta_p)$ corresponding to $C_{1-k}$. How do we construct this extension? Ribet managed to do so using modular forms. For example: $p=691$ divides $B_{12}$ and the explicit formula gives a congruence between $E_{12}$ and $\Delta$ modulo $p=691$; the desired extension of $\mathbb{Q}(\zeta_p)$ is then constructed using the Galois representation attached to $\Delta$.


TopAutomorphic representations on $GL_2(\mathbb{A}^\infty)$

It begins to reveal the remarkable nature of modular/automorphic forms when you add a little bit representation theory.

Definition 1 Let $F$ be a number field and $\mathbb{A}_F$ be the ring of adeles of $F$. The group $GL_2(\mathbb{A}_F)$ is defined to be the restricted direct product $\prod_v'GL_2(F_v)$ with respect to the open compact subgroups $\{GL_2(\mathcal{O}_v),v\nmid \infty\}$. Notice as an abstract group $GL_2(\mathbb{A}_F)$ is simply the group of invertible matrices with entries in $\mathbb{A}_F$. We now consider $F=\mathbb{Q}$ and write $\mathbb{A}=\mathbb{A}_\mathbb{Q}$ for short.
Definition 2 Let $\mathcal{H}^{\pm}=\{z\in \mathbb{C}: \Im(c)\ne0\}$. Fix $k\ge1$ an integer and $U\subseteq GL(\mathbb{A}^\infty)$ be an open compact subgroup. A modular form of level $U$ is a function $f: GL_2(\mathbb{A}^\infty)\times\mathcal{H}^{\pm}\rightarrow \mathbb{C}$ satisfying:
  1. For any $g\in GL_2(\mathbb{A}^\infty)$, $\tau\mapsto f(g,\tau)$ is holomorphic.
  2. For any $\gamma\in GL_2(\mathbb{Q})$, $f(\gamma (g,\tau))=\det(\gamma)^{-1}j(\gamma,\tau)^kf(g,\tau)$ (Notice $GL_2(\mathbb{Q})$ acts on both factors from the left).
  3. For any $u\in U$, $f(gu,\tau)=f(g,\tau)$.
  4. For any $g\in GL_2(\mathbb{A}^\infty)$, $\tau\mapsto f(g,\tau)$ is holomorphic at $\infty$ (i.e., tends a finite limit when $\tau\rightarrow\infty$).

If further $\tau\rightarrow f(g,\tau)$ vanishes at $\infty$, we say that $f$ is a cusp form of level $U$.

This complicated definition generalizes the classical notion of modular forms (the advantage is that it puts the representation theory in scope as we consider all levels $U$).

Theorem 1 Suppose the open compact subgroup $U\subseteq GL_2(\hat{\mathbb{Z}})$ satisfies $\det(U)=\hat{\mathbb{Z}}^\times$. Then $\Gamma=U\cap GL_2(\mathbb{Q})^+$ is a congruence subgroup of $SL_2(\mathbb{Z})$ and $$M_k(\Gamma)\cong M_k(U),\quad S_k(\Gamma)\cong M_k(U).$$
Remark 1 Any open compact subgroup can be conjugate into $GL_2(\hat{\mathbb{Z}})$. But the special shape of $U$ is required for the theorem to hold.

To prove the theorem, we need the following

Lemma 1
  1. $SL_2(\mathbb{Z})\rightarrow SL_2(\mathbb{Z}/N \mathbb{Z})$ is surjective.
  2. $GL_2(\mathbb{A}^\infty)=GL_2(\mathbb{Q})GL_2(\hat{\mathbb{Z}})$.
  3. $SL_2(\mathbb{A}^\infty)=SL_2(\mathbb{Q})SL_2(\hat{\mathbb{Z}})$.
  1. Pick an arbitrary lift $Y$. We would like a matrix $\gamma$ such that $\det(\gamma)=1$ and $\gamma\equiv Y\pmod{N}$. By multiplying $Y$ on the left and right by $SL_2(\mathbb{Z})$, we may assume $Y\equiv\diag\{x,y\}$. Then $xy\equiv1\pmod{N}$ and the matrix $$\gamma=
  x & -(1-xy) \\
  1-xy & y(2-xy)
\end{bmatrix}$$ is desired.
  2. Using $\mathbb{Q}\cap \hat{\mathbb{Z}}= \mathbb{Z}$ and $\mathbb{Q}+ \hat{\mathbb{Z}}= \mathbb{A}^\infty$, we know that there is a bijection between free $\mathbb{Z}$-modules in $\mathbb{Q}^n$ of rank $n $ and free $\hat{\mathbb{Z}}$module of $(\mathbb{A}^\infty)^n$ of rank $n $ given by $M\mapsto \hat M=M\otimes \hat{\mathbb{Z}}$, $\hat M\mapsto M=\hat M\cap \mathbb{Q}^n$. Consequently, it gives a bijection $GL_2(\mathbb{Z})\backslash GL_2(\mathbb{Q})\cong GL_2(\hat{\mathbb{Z}})\backslash GL_2(\mathbb{A}^\infty)$. In particular the desired result follows.
  3. Let $g\in SL_2(\mathbb{A}^\infty)$. Use part (b) to write $g=\gamma\cdot k$, where $\gamma\in GL_2(\mathbb{Q})$, $k\in GL_2(\hat{\mathbb{Z}})$. Taking determinants, we know $\det (\gamma)=\mathbb{Q}^\times\cap \hat{\mathbb{Z}} ^\times=\{\pm1\}$. The claim follows from changing $\gamma$ by an element of $GL_2(\mathbb{Z})$ if necessary.
Remark 2 (b) and (c) is special to $GL_n$ and $SL_n$ and $F=\mathbb{Q}$ (so called class number one). (a) for $SL_n$ is a consequence of strong approximation of $SL_n$ (Here we will use (a) to prove the strong approximation for $SL_2$).
Theorem 2 (Strong approximation for $SL_2$) $SL_2(\mathbb{Q})$ is dense in $SL_2(\mathbb{A}^\infty)$.
Proof Let $U\subseteq SL_2(\mathbb{A}^\infty)$ be any open compact subgroup. We will show that $SL_2(\mathbb{Q})\cdot U=SL_2(\mathbb{A}^\infty)$. We may assume that $U\supseteq \hat\Gamma (N)$, where $\hat\Gamma(N)=\ker(SL_2(\hat{\mathbb{Z}})\twoheadrightarrow SL_2(\mathbb{Z}/N \mathbb{Z}))$. By (c) of the previous lemma, we can write $g\in SL_2(\mathbb{A}^\infty)$ as $g=\gamma k$, where $\gamma\in SL_2(\mathbb{Q})$ and $k\in SL_2(\hat{\mathbb{Z}})$. By (a) of the previous lemma, we can find $l\in SL_2(\mathbb{Z})$ such that $l\equiv k\pmod{N}$. Then $g=(gl)(l^{-1}k)\in SL_2(\mathbb{Q})\cdot U$.
Corollary 1 Let $U\subseteq GL_2(\hat{\mathbb{Z}})$ be an open compact subgroup such that $\det (U)=\hat{\mathbb{Z}}^\times$. Then
  1. $GL_2(\mathbb{Q})\cdot(U\times GL_2(\mathbb{R})^+)=GL_2(\mathbb{A})$
  2. $GL_2(\mathbb{Q})\cdot(U\times\mathcal{H}^{\pm})=GL_2(\mathbb{A}^\infty)\times\mathcal{H}^{\pm}$.
  1. It follows from the $\det(GL_2(\mathbb{Q})\cdot (U\times GL_2(\mathbb{R})^+)=\mathbb{Q}^\times(\hat{\mathbb{Z}}^\times\cdot \mathbb{R}_{>0})=\mathbb{A}^\times$ and the strong approximation of $SL_2$.
  2. It follows from the first part because $GL_2(\mathbb{R})$ acts transitively on $\mathcal{H}^{\pm}$.
Proof (Theorem 1) The bijection is constructed as $f\mapsto F(\tau)=f(1,\tau)$ and $F\mapsto f(g,\tau)=\det(\gamma)^{2-k}F|_k[\gamma](\tau)$, where $\gamma\in GL_2(\mathbb{Q})$ such that $\gamma(g,\tau)\in U\times \mathcal{H}$, whose existence is assured by the previous corollary. (One needs to check the latter is well-defined and $f(g,\tau)$ thus defined is a modular form of level $U$).
Remark 3 Most common choice of $U$: $$U_1(N)=\left\{
  a & b \\
  c & d
\end{bmatrix}\in GL_2(\hat{\mathbb{Z}}): c\equiv0\pmod{N}, d\equiv 1\pmod{N}\right\}.$$ So $\det (U_1(N))=\hat{\mathbb{Z}}^\times$ and $U_1(N)\cap GL_2(\mathbb{Q})^+=\Gamma_1(N)$.

The most interesting thing is that the group $GL_2(\mathbb{A}^\infty)$ acts on both of the spaces $$\mathcal{M}_k=\varinjlim_{U \text{ open compact}} M_k(U),\quad \mathcal{S}_k=\varinjlim_{U \text{ open compact}} S_k(U)$$ by $(g.f)(h,\tau)=f(hg,\tau)$ (so $g.f\in M_k(g^{-1}Ug)$ if $f\in M_k(U)$).

Definition 3 An automorphic representation of weight $k$ is an irreducible representation $\pi$ of $GL_2(\mathbb{A}^\infty)$ which is isomorphic to a subquotient of $\mathcal{M}_k$. Similarly, a cuspidal automorphic representation is a subquotient of $\mathcal{S}_k$.

TopRepresentations of locally profinite groups

A representation of $GL_2(\mathbb{A}^\infty)$ means more precisely the following.

Definition 4 A representation of a locally profinite group $G$ is a pair $(\pi, U)$, where $U$ is a $\mathbb{C}$-vector space and $\pi: G\rightarrow\Aut(U)$ is a homomorphism. A representation $\pi$ is called smooth if for any $v\in U$, there exists an open compact subgroup $K\subseteq G$ such that $v\in U^K$. It is called admissible if it is smooth and $U^K$ is finite dimensional for any open compact subgroup $K$.
Remark 4 $\mathbf{Rep}(G)$, the category of smooth representations of $G$, is abelian, and in particular, the notion of subquotient is well defined. Moreover, any subquotient of an admissible representation is still admissible.
Remark 5 By definition, one have $\mathcal{M}_K^U=M_k(U)$ and $\mathcal{S}_k^U=S_k(U)$. One can show that $M_k(U)$ and $S_k(U)$ is always finite dimensional (a special case follows from Theorem 1). Therefore an automorphic representation $\pi$ of $GL_2(\mathbb{A}^\infty)$, as a subquotient of the admissible representation $\mathcal{M}_k$, is always admissible.


Remark 6
  1. If $G$ is compact and $(\pi,V)$ is an irreducible, smooth representation, then $\pi$ is finite dimensional, as $\ker \pi$ contains an open normal subgroup of $G$ and any open subgroup of $G$ is of finite index.
  2. In general, if $(\pi,V)$ is smooth and $K\subseteq G$ an open compact subgroup. Then for any $v\in V$, $\mathbb{C}[K]\cdot v\subseteq V$ is finite dimensional and semisimple as a representation of $K$.
  3. If $\sigma$ is a smooth irreducible representation of $K$. We write $V[\sigma]\subseteq V$ for the span of all $K$-subspaces of $V$ which are isomorphic to $\sigma$. Then $V=\bigoplus_\sigma V[\sigma]$, where $\sigma$ runs over all smooth irreducible representation of $K$. Moreover, $V$ is admissible if and only if $V[\sigma]$ is finite dimensional for any $\sigma$ (a condition that does not require to check for every $K$).
Definition 5 Let $C_c^\infty(G)=\{f: G\rightarrow \mathbb{C}: \text{ compactly supported and locally constant}\}$ (here $c$ stands for compactly supported and $\infty$ stands for locally constant). This becomes a smooth representation of $G$ under left and right translations: $$L(g)(f)(x)=f(g^{-1}x),\quad R(g)(f)(x)=f(xg).$$ For any $f\in C_c^\infty$, one can write it as $$f=\sum_{i=1}^n\alpha_i \mathbf{1}_{K g_i K}=\sum\beta_i \mathbf{1}_{g_iK}=\sum\gamma_i\mathbf{1}_{Kg_i},$$ where $K$ is some compact open subgroup depending on $f$.
Definition 6 As $G$ is a locally compact group, there exists a Haar measure $C_c^\infty(G)\rightarrow \mathbb{C}$, denoted by $f\mapsto \int_G f(g)dg$ satisfying
  1. If $f$ takes positive real values then $\int_Gfdg\ge0$.
  2. Left-invariance: $\int_Gf(gh)dh=\int_Gf(h)dh$.

The functional $\int_G \cdot dg $ is unique up to positive real multiple (and we choose one). We will assume that $G$ is unimodular, i.e., the Haar measure is both left and right invariant, $$\int_G f(hg)=dh=\int_G f(h)dh.$$

Example 4 The group $G=GL_2(\mathbb{Q}_p)$ and $G=GL_2(\mathbb{A}^\infty)$ are unimodular. In general, if $\mathcal{G}$ is any reductive group over $\mathbb{Q}_p$, then $G=\mathcal{G}(\mathbb{Q}_p)$ is also unimodular.
Remark 7 Using the Haar measure, $C_c^\infty(G)$ becomes an associated algebra under $$(f_1\cdot f_2)(g)=\int_{G} f_1(x)f_2(g^{-1}x)dx.$$ If $(\pi ,V)$ is a smooth representation of $G$, then $V$ becomes a module over the algebra $C_c^\infty(G)$ given by $$\pi(f)(v)=\int_G f(g)\pi(g)(v)dg.$$ If $f=\sum_{i}\alpha_i \mathbf{1}_{g_iK}$, then $\pi(f)(v)=\sum_i \alpha_i \mu(K) \pi(g_i)(v)$, a finite combination of the linear operators induced from $\pi$.
Remark 8 A representation of $G$ is the same thing as a $\mathbb{C}[G]$-module and a smooth representation of $G$ is the same thing as a $C_c^\infty(G)$-module.
Definition 7 Now fix an open compact subgroup $K\subseteq G$ and normalize the Haar measure such that $\mu(K)=1$. We define the Hecke algebra $\mathcal{H}(G,K)\subseteq C_c^\infty(G)$ to be the space of compactly supported, locally constant functions on $G$ which are $K$-bi-invariant. In particular, $C_c^\infty(G)=\bigcup_K \mathcal{H}(G,K)$.
Lemma 2 Let $e_K=\mathbf{1}_{K}$. Then for any $f\in C_c^\infty(G)$, $f\in \mathcal{H}(G,K)$ if and only if $e_k f=f=f e_K$.

Since $e_k\cdot e_k=e_k$, it follows that $\mathcal{H}(G,K)=e_K C_c^\infty(G) e_K$ is a subalgebra of $C_c^\infty(G)$ with the unit $e_K$.

Remark 9 $\mathcal{H}(G,K)$ has a basis consisting of $\mathbf{1}_{K gK}$ for $g\in K\backslash G/ K$. But the multiplication law is complicated even for simple subgroup $K\subseteq G$.
Theorem 3
  1. Let $(\pi, V)$ be a smooth representation of $G$. Then for any $f\in \mathcal{H}(G,K)$, $\pi(f)V\subseteq V^K$. Therefore $V^K$ is a $\mathcal{H}(G,K)$-module.
  2. Let $(\pi, V)$ be an admissible irreducible representation of $G$. Then either $V^K=0$ or $V^K$ is an irreducible $\mathcal{H}(G,K)$-module. Moreover, if for $\pi_1, \pi_2$ such that $\pi_i^K\ne0$, we have $\pi_1\cong \pi_2$ if and only if $\pi_1^K\cong \pi_2^K$ as $\mathcal{H}(G,K)$-modules.
Remark 10 The above bijection $\pi\mapsto \pi^K$ when $\pi^K\ne0$ is not quite functorial (e.g., doesn't preserve extensions). For example, in the category of smooth representations of $GL_2(\mathbb{Q}_p)$, any extension of the trivial representation by itself splits. But the module $\mathbb{C}^2$ where $T_p$ acts trivially and $S_p$ acts as $\left(\begin{smallmatrix} 1& 1\\0 &1\end{smallmatrix}\right)$ is a nontrivial extension of the trivial module by itself.

Now let us consider the case $G=GL_2(\mathbb{Q}_p)$ and $K=GL_2(\mathbb{Z}_p)$. Every compact subgroup $L\subseteq G$ is contained in a $G$-conjugate of $K$and $K$ is a maximal compact subgroup.

Definition 8 We say an admissible irreducible representation $(\pi,V)$ is unramified if $\pi^K\ne0$.
Theorem 4 Let $T_p=\mathbf{1}_{K
    1 & 0 \\
    0 & p
}$ and $S_p=\mathbf{1}_{K
  p & 0 \\
  0 &p
\end{smallmatrix}\right)K}$. Then $\mathcal{H}(G,K)\cong \mathbb{C}[T_p, S_p^{\pm}]$. In particular, $\mathcal{H}(G,K)$ is commutative.
Remark 11 This follows from the Satake isomorphism for $GL_2$ and we will prove a general version later.
Remark 12 The commutativity is very special to the maximal compact subgroup $K=GL_2(\mathbb{Z}_p)$. The Hecke algebra $\mathcal{H}(G,K)$ can be highly non-commutative for other open compact subgroups $K$.
Corollary 2 If $\pi$ is an unramified representation of $G$. Then $\pi^K$ is 1-dimensional, determined by the eigenvalues of $T_p$ and $S_p$.
Remark 13 This does us good because in general $\pi$ is infinite dimensional!

TopBack to automorphic representations on $GL_2(\mathbb{A}^\infty)$

When $p\nmid N$, we can write $U_1(N)=U_1(N)^p\times GL_2(\mathbb{Z}_p)\subseteq GL_2(\hat{\mathbb{Z}})$. Therefore $$\mathcal{S}_k^{U_1(N)}=(\mathcal{S}^{U_1(N)^p})^{GL_2(\mathbb{Z}_p)}\subseteq(\mathcal{S}_K))^{U_1(N)^p},$$ the latter admits an action of $GL_2(\mathbb{Z}_p)$, so $\mathcal{S}_k^{U_1(N)}$ is a $\mathcal{H}(G_p,K_p)$-module, where $G_p=GL_2(\mathbb{Q}_p)$ and $K_p=GL_2(\mathbb{Z}_P)$.

Definition 9 The operator $T_p\in \mathcal{H}(G_p,K_p)$ then induces an operator, still denoted by $T_p$, acting on $S_k(\Gamma_1(N))=S_k(U_1(N))$.
Example 5 When $N=1$, this adelically defined operator agrees with the classical Hecke operators $T_p$ acting on $S_k(SL_2(\mathbb{Z}))\cong S_k(GL_2(\hat{\mathbb{Z}}))$. Indeed, decompose $$K_p
  1 & 0 \\
  0 & p
  p & -i \\
  0 & 1
  1 & 0 \\
  0 & p
\end{smallmatrix}\right)K_p,$$ and call these representatives $\beta_0,\ldots \beta_{p-1}$ and $\beta_p$ viewed as elements in $GL_2(\mathbb{Q}_p)\subseteq GL_2(\mathbb{A}^\infty)$. Starting from $F\in S_k(SL_2(\mathbb{Z}))$, we want to show that $T_p^\text{adelic}(F)=T_p(F)$. Notice $$T_p^\text{adelic}(F)(\tau)=\int_{K_p
 \left( \begin{smallmatrix}
    1 & 0 \\
    0 & p
  \end{smallmatrix}\right)K_p}f(g,\tau)dg=\sum_{i=0}^pf(\beta_i,\tau).$$ Define $\gamma_i$ as the $\beta_i$ viewed as elements in $GL_2(\mathbb{Q})\subseteq GL_2(\mathbb{A}^\infty)$. Then $\gamma_p^{-1}\beta_p$ has entry 1 at the prime $p$ and $
  1 & 0 \\
  0 & p^{-1}
\end{smallmatrix}\right)$ at other primes $q\ne p$, hence is an elements of $GL_2(\hat{\mathbb{Z}})$. Thus $$\sum_{i=0}^pf(\beta_i,\tau)=\sum_i\det(\gamma_i^{-1})j(\gamma_i^{-1},\tau)^{-k}F(\gamma_i^{-1}\tau)=T_p(F)$$ as desired when you expand the terms.

Similarly, $S_p\in \mathcal{H}(G_p,K_p)$ induces a operator $S_p$ on $S_k(SL_2(\mathbb{Z}))$. One can compute that $S_pF=p^{k-2}F$.

Since $\mathcal{S}_k^{GL_2(\hat{\mathbb{Z}}^p)}$ is a semisimple representation of $GL_2(\mathbb{Q}_p)$ (Corollary 3), we can write $$\mathcal{S}_k^{GL_2(\hat{\mathbb{Z}}^p)}=\bigoplus_{i=1}^\infty \pi_i,$$ where $\pi_i$'s are the irreducibles. This implies that $$S_k(SL_2(\mathbb{Z}))=\bigoplus\pi_i^{K_p}.$$ After reordering, we can assume $\pi_i^{K_p}\ne0$ (hence is 1-dimensional) if and only if $i=1,\ldots, r$. The isomorphism classes of $\pi_i$, $i=1,\ldots, r$ are then determined by the isomorphism class of $\pi_i^{K_p}$ as a $\mathcal{H}(G_p,K_p)$-module, or even better, by the eigenvalues of $T_p$ and $S_p$ ($=p^{k-2}$).


Definition 10 A smooth representation of a locally profinite group $G$ is called unitary if there exists a Hermitian positive definite inner product $\langle\cdot, \cdot\rangle: V\times V\rightarrow \mathbb{C}$ that is $G$-invariant. When $G$ is compact (e.g., finite), every irreducible representation is unitary.
Lemma 3 If $(\pi, V)$ is unitary and admissible representation of $G$, then it is semisimple. Namely $$\pi \cong\bigoplus_{i\in I} \pi_i,$$ where each $\pi_i$ is irreducible and admissible.
Proof By Zorn's lemma, it suffices to show that any $G$-invariant subspace of $W\subseteq V$ has $G$-invariant complement, namely $V=W\oplus W'$. We will show that we can take $W'{}=W^\perp=\{v\in V: \langle v,w\rangle=0, \forall w\in W\}$. By positive definiteness, $W\cap W^\perp=0$. It remains to show that $W+W^\perp=V$. Choose $K\subseteq G$ an open compact subgroup. Recall that (Remark 6) $$V=\bigoplus_\sigma V[\sigma].$$ We then need to show that $V[\sigma]\subseteq W+W^\perp$. Because $V[\sigma]$ is finite dimensional, we have $V[\sigma]=W[\sigma]\oplus (V[\sigma]\cap W[\sigma]^\perp)$. For any $\tau$ a smooth irreducible of $K$, $W[\tau]$ is perpendicular to $V[\sigma]\cap W[\sigma]^\perp$ (check this according $\tau=\sigma$ or $\tau\ne\sigma$). We see that $V[\sigma]\cap W[\sigma]^\perp\subseteq W^\perp$, since $W=\bigoplus W[\tau]$. It follows that $V[\sigma]\subseteq W+W^\perp$.
Corollary 3 If $k\ge1$, then $\mathcal{S}_k$ is a semisimple representation of $GL_2(\mathbb{A}^\infty)$. So we can write $\mathcal{S}_k=\bigoplus \pi$, where $\pi$ runs over all cuspidal automorphic representations of $GL_2(\mathbb{A}^\infty)$.
Proof Want to show that $\mathcal{S}_k$ is a unitary representation. This is not quite true but there exists a character $\psi: GL_2(\mathbb{A}^\infty)\rightarrow \mathbb{C}^\times$ such that $\mathcal{S}_k\otimes \psi$ is unitary. The unitary structure is a generalization of the Peterson inner product which we will talk about next time. This is enough to show that $\mathcal{S}_k$ is semisimple by untwisting the direct sum decomposition of $\mathcal{S}_k\otimes \psi$.
Definition 11 Suppose we are given a finite set of primes $S$ and for any $p$, an irreducible admissible representation $\pi_p$ of $G_p$. Suppose for $p\not\in S$, $\pi_p$ is unramified. So $\pi_p^{K_p}$ is one dimensional and we choose a nonzero vector $x_p\in \pi_p^{K_p}$. Let $T\supseteq S$ be a finite set of primes.
  • We define $V_T=\bigotimes_{p\in T} \pi_p$. Then $G_T=\prod_{p\in T}G_p\times\prod_{p\not\in T} K_p$ acts on $V_T$ (by the trivial action if $p\not\in T$).
  • If $T\subseteq T'$, we define a map $$V_T\rightarrow V_{T'},\quad \bigotimes_{p\in T}v_p\mapsto \bigotimes_{p\in T}v_p\bigotimes_{p\in T'-T } x_p.$$ It is compatible with the action of $G_T\hookrightarrow G_{T'}$.
  • We define the restricted tensor product of $\pi_p$ to be the $GL_2(\mathbb{A}^\infty)$-representation $$\bigotimes_p{}' \pi_p=\varinjlim_T V_T,$$ where $T$ runs over all finite set of primes containing $S$.
Proposition 1
  1. $\bigotimes_p' \pi_p$ is an irreducible admissible representation of $GL_2(\mathbb{A}^\infty)$, which only depends on $\pi_p$ and not on $S$ or $x_p$'s.
  2. If $\pi$ is any irreducible admissible representation of $GL_2(\mathbb{A}^\infty)$, there exists $\pi_p$'s such that $\pi=\bigotimes_p' \pi_p$. Moreover, the isomorphism class of the $\pi_p$'s are uniquely determined.
Proof The proof is purely algebraic, see Flath in Corvallis I. We do remark that the some holds if $GL_2$ is replaced by any restricted direct product $\prod_{i\in I}'G_i$ with respect to $(G_i,K_i)_{i\in I}$ such that for almost all $i$, the Hecke algebra $\mathcal{H}(G_i, K_i)$ is commutative (e.g., $G(\mathbb{A}^\infty)$ for any reductive group $G$).

We now can describe the summands of $\mathcal{S}_k$ in classical terms.

Proposition 2
  1. Let $N\ge1$ and $p\nmid N$ a prime , then $T_p$ acts semisimply on $S_k(\Gamma_1(N))$ and the $T_p$, $T_q$ commute for $q\nmid N$ a prime.
  2. If $f\in S_k(\Gamma_1(N))$ is an eigenform for all $T_p$, $p\nmid N$, then the submodule of $\mathcal{S}_k$ generated by $f$ is irreducible. Conversely, any irreducible submodules of $\mathcal{S}_k$ is obtained in this way.
  1. Let $\pi_1,\ldots \pi_r$ be the irreducible submodules of $\mathcal{S}_k$ such that $\pi_i^{U_1(N)}\ne0$ (there are only finitely many by the admissibility of $\mathcal{S}_k$. As modules for $\mathbb{C}[T_p]$, $$S_k(\Gamma_1(N))\cong\bigoplus_{i=1}^r\left(\bigotimes_{q\ne p} \pi_{i,q}^{U_1(N)^p}\bigotimes \pi_{p}^{GL_2(\mathbb{Z}_p)}\right).$$ Then $T_p$ acts on the first factor trivially and acts on the second factor (which is 1-dimension) in the usual way. It follows that the action of $T_p$ is semisimple and $T_p$, $T_q$ commute.
  2. We Need some nontrivial global information:
Theorem 5
  1. (Multiplicity one) If $V,V'\subseteq \mathcal{S}_k$ are irreducible submodules such that $V\cong V'$, then $V=V'$. Namely, $\mathcal{S}_k$ decomposes with multiplicity one.
  2. (Strong multiplicity one). If $V=\bigotimes_p'\pi_p$ and $V=\bigotimes_p'\pi_p'$ are cuspidal automorphic representations of $GL_2(\mathbb{A}^\infty)$ of weight $k$ such that for almost all $p$ (e.g., all unramified primes), $V_p\cong V'_p$, then $V\cong V'$.
Remark 14 This theorem is still true for $GL_n$ but not for general reductive group. So there is actual work to be done and we will not talk about the proof.
Remark 15 From the view of automorphic representations, the difficulty of the study of Hecke operators at $p\mid N$ in classical modular forms are accounted by the more complicated representation theory of the ramified representations of $GL_2(\mathbb{Q}_p)$.

Let $V\subseteq \mathcal{S}_k$ be the submodule generated by $f$. By the semisimplicity, we can write $V=V_1\oplus\cdots V_r$, where $V_i$'s are irreducible. Notice that $V_{i,p}\cong V_{j,p}$ because these are unramified representations and their isomorphism class is determined by eigenvalue of $T_p$ on $f$. Now we apply strong multiplicity to obtain $V_i\cong V_j$ and the multiplicity one implies that $r=1$.

For the converse, we need some nontrivial local information:

Theorem 6 Let $m\ge0$ and define $$K_1(p^m)=\{\left(
  a & b\\
 c & d
\end{smallmatrix}\right)\in GL_2(\mathbb{Z}_p): c\equiv0(p^m), d\equiv1(p^m)\}.$$ If $\pi$ is any irreducible admissible representations of $GL_2(\mathbb{A}^\infty)$, then there exists $m\ge0$ such that $\pi^{K_1(p^m)}\ne0$. The smallest $m$ is called the conductor of $\pi$.
Remark 16 One should require such that $\pi_p$ is infinite dimensional (generic in general terminology), this is automatically satisfied when $\pi$ is a cuspidal automorphic representation; otherwise $\pi_p$ can be (and must be, by Schur's lemma) of the form $\chi\circ\det$.

By this theorem, there exists $N\ge1$ such that $\pi^{U_1(N)}\ne0$ (do it at one prime a time). If $\pi$ is a cuspidal automorphic representation, then we can chose $f\in \pi^{U_1(N)}\subseteq \mathcal{S}_k^{U_1(N)}\cong S_k(\Gamma_1(N))$. Then $f$ is a $T_p$-eigenvector for all $p\nmid N$ and $f$ generates $\pi$.

Remark 17 If we don't have this theorem, we can certainly obtain the similar result for $\Gamma(N)$ ($\Gamma(N)$ exhausts all open compact subgroups, but $\Gamma_1(N)$ doesn't). The point is that modular forms on $\Gamma_1(N)$ are much simpler than those on $\Gamma(N)$ and it suffices to look at modular forms on $\Gamma_1(N)$ from the view of automorphic representations.

TopGeneral automorphic representations on $GL_2(\mathbb{A})$

Reference on this section:

  • Gelbart, Automorphic forms on adeles groups
  • Deligne, Formes modulaires et representations de GL(2)

Now we are going to also incorporate the representation theory at $\infty$ and enlarge the notion of automorphic forms and representations on $GL(2)$. We write $G=GL_2(\mathbb{R})$, $\mathfrak{g}=\Lie G=M_2(\mathbb{R})$, $K^0=SO(2)$ and $K=O(2)=K\coprod \left(
    0 & 1\\
    1 & 0
  \end{smallmatrix}\right)K$. Then $K$ is a maximal compact subgroup of $G$ and write $\mathfrak{k}=\Lie K$.

Definition 12 We define $$C^\infty(G)=\{f: G\rightarrow \mathbb{C}: \text{smooth}\},$$ $$C_c^\infty(G)=\{ f: G\rightarrow \mathbb{C}: \text{ smooth compactly supported}\}.$$ Then $G$ acts on the left on $C^\infty(G)$ (or $C_c^\infty(G)$) by right translation and $\mathfrak{g}$ acts on $G$ by $$(Xf)(x)=\frac{d}{dt}f(\exp(tX))|_{t=0}, \quad f\in C^\infty(G), X\in \mathfrak{g}.$$ This makes $C^\infty(G)$ (or $C_c^\infty(G)$) into a representation of $\mathfrak{g}$ and hence extends to a representation of $U(\mathfrak{g})$, the universal enveloping algebra of $\mathfrak{g}$. By linearity, $U(\mathfrak{g}_\mathbb{C})$ also acts on $C^\infty(G)$ (or $C_c^\infty(G))$, where $\mathfrak{g}_\mathbb{C}=\mathfrak{g}\otimes_\mathbb{R} \mathbb{C}$.
Example 6The vector $w=\left(
  0 & 1\\
  -1 & 0
\end{smallmatrix}\right)$ spans $\mathfrak{k}=\Lie K$. The vectors $X_+=\left(
  1 & i\\
  i & -1
\end{smallmatrix}\right)$ and $X_-=\left(
  1 & -i\\
  -i & -1
\end{smallmatrix}\right)$ are the eigenvectors for the adjoint action of $K$ or $\mathfrak{k}$ on $\mathfrak{g}_\mathbb{C}$ with non-zero eigenvalue. We have the commutant relations $$[w,X_+]=2iX_+,\quad [w,X_-]=-2iX_-,\quad [X_+,X_-]=-4iw.$$ Let $Z(\mathfrak{g})\subseteq U(\mathfrak{g})$ be the center of the universal enveloping algebra. One can show that $Z(\mathfrak{g})=\mathbb{C}[z,\Delta]$, where $z=\left(
  1 & 0\\
  0 &1
\end{smallmatrix}\right)$ and $\Delta=-\frac{1}{4}(X_-X_+-2iw-w^2)$ is the Casmir operator.


Proposition 3 Let $f:\mathcal{H}\rightarrow \mathbb{C}$ be a smooth function. We associate to $f$ a function $$\phi: GL_2(\mathbb{R})^+\rightarrow \mathbb{C},\quad \phi(g)=\det(g)f(gi)j(g,i)^{-k},$$ where $k\ge1$. Then
  1. $\phi(\left(\begin{smallmatrix} z & 0 \\ 0 & z\end{smallmatrix}\right)gr_\theta )=z^{2-k}e^{ik\theta}\phi(g)$ for any $r_\theta \in K^0=SO(2)$, $z\in \mathbb{R}^\times$.
  2. $f$ is holomorphic if and only if $X_-\phi=0$.
Proof We use the Iwasawa decomposition $GL_2(\mathbb{R})^+=ZNAK^0$, so any $g\in GL_2(\mathbb{R})^+$ has a unique decomposition $$g= \left(\begin{smallmatrix} z & 0 \\ 0 & z\end{smallmatrix}\right) \left(\begin{smallmatrix}1 & x \\ 0 &1\end{smallmatrix}\right) \left(\begin{smallmatrix}y^{1/2} & 0 \\ 0 & y^{-1/2}\end{smallmatrix}\right)  r_\theta.$$ Using the fact that $j(g,i)=ze^{-i\theta}y^{-1/2}$, we see that $\phi(g)=z^{2-k}f(x+iy)y^{k/2}e^{ik\theta}$ and the first part follows. For the second part, one compute the differential operator $$X_-=e^{-2i\theta}\left(-2iy\frac{\partial}{\partial x}+2y \frac{\partial}{\partial y}+i\frac{\partial}{\partial \theta}\right)$$ and find out that $X_- \phi=0$ if and only if $i\partial f/\partial x=\partial f/\partial y$.
Definition 13 Let $f: GL_2(\mathbb{A}^\infty)\times\mathcal{H}^{\pm}\rightarrow \mathbb{C}\in \mathcal{M}_k$, we associate to $f$ a function $\phi: GL_2(\mathbb{A})\rightarrow \mathbb{C}$ as above. Then it satisfies that (c.f., Definition 2)
  1. For any $g^\infty\in GL_2(\mathbb{A}^\infty)$, $g_\infty\mapsto\phi(g^\infty g_\infty)$ is holomorphic.
  2. There exists open compact subgroup $U\subseteq GL_2(\mathbb{A}^\infty)$ such that for any $g\in GL_2(\mathbb{A})$ and $u\in U$, $\phi(gu)=\phi(g)$.
  3. For any $g\in GL_2(\mathbb{A})$ and $\gamma\in GL_2(\mathbb{Q})$, $\phi(\gamma g)=\phi(g)$.
  4. For any $g\in GL_2(\mathbb{A}^\infty)$ and $z\in \mathbb{R}^\times$, $\phi(g \left(\begin{smallmatrix}z & 0\\0 & z\end{smallmatrix}\right) )=z^{2-k}\phi(g)$.
  5. For any $g\in GL_2(\mathbb{A})$, $\phi(g r_\theta)=e^{ik\theta}\phi(g)$.
  6. $X_-\phi=0$.
  7. For any $g^\infty\in GL_2(\mathbb{A}^\infty)$, there exists $N,C>0$ such that $$|\phi(g^\infty g_\infty)|\le  C||g_\infty||^N$$ for any $g_\infty\in GL_2(\mathbb{R})$, where for $g=\left(\begin{smallmatrix}a & b\\c &d\end{smallmatrix}\right)\in GL_2(\mathbb{R})$, $||g||=\Tr {}^tgg+\Tr {}^tg^{-1}g^{-1}=(a^2+b^2+c^2+d^2)(1+\det(g^{-2}))$.
Proposition 4 The above association $f\mapsto \phi$ gives an isomorphism of $GL_2(\mathbb{A}^\infty)$-modules $\mathcal{M}_k$ and the functions $\phi$ satisfying the above 7 conditions. This restricts to an isomorphism between $\mathcal{S}_k$ and the $\phi$'s which also satisfy $$\int_{\mathbb{Q}\backslash \mathbb{A}}\phi( \left(\begin{smallmatrix}1 & x\\ 0 &1\end{smallmatrix}\right))g)dx=0,\quad\forall g\in GL_2(\mathbb{A}).$$
Proof It is easy to check the isomorphism by construction. To match the cuspidality condition, it suffices to check on $g$ of the form $\left(\begin{smallmatrix}a & 0\\ 0 &a^{-1}\end{smallmatrix}\right)$, $a\in \mathbb{R}_{>0}$. Using $\mathbb{Z}\backslash \mathbb{R}\cong \mathbb{Q}\backslash \mathbb{A}/\hat{\mathbb{Z}}$ and the Cauchy integral formula, we see that $$a^k\int_0^1f(1,x+a^2i)dx=a^ka_0,$$ where $f(1,\tau)=\sum_{n\ge0} a_nq^n$.
Remark 18 The best way to define the Peterson inner product (the unitary structure on $\mathcal{S}_k$ up to twist): for $\phi_i\in GL_2(\mathbb{Q})\backslash GL_2(\mathbb{A})\rightarrow \mathbb{C}$, we define $$\langle\phi_1,\phi_2\rangle=\int_{\mathbb{A}^\times GL_2(\mathbb{Q}) \backslash GL_2(\mathbb{A})}\phi_1(g)\overline{\phi_2(g)}|\det(g)|^{2-k}dg.$$ The domain indeed has finite volume.
Definition 14 The space $\mathcal{A}$ of automorphic forms on $GL_2$ is the space of functions $\phi: GL_2(\mathbb{A})\rightarrow \mathbb{C}$ satisfying:
  1. Left invariant under $GL_2(\mathbb{Q})$.
  2. $g_\infty\mapsto \phi(g^\infty g_\infty)$ is smooth.
  3. Right $K=O(2)$-finiteness.
  4. Right invariant under some open compact subgroup $U\subseteq GL_2(\mathbb{A}^\infty)$.
  5. $Z(\mathfrak{g})$-finiteness (generalizing the holomorphy condition).
  6. Moderate growth: $|\phi(g^\infty g_\infty)|\le C||g_\infty||^N$ (generalizing the holomorphy condition at infinity )

The space of cuspidal automorphic forms $\mathcal{A}_0\subseteq \mathcal{A}$ is defined by the further condition $$\int_{\mathbb{Q}\backslash \mathbb{A}}\phi(\left(\begin{smallmatrix}1 & x \\ 0 &1\end{smallmatrix}\right) g)dx=0,\quad g\in GL_2(\mathbb{A}).$$

Remark 19 $\mathcal{A}$ becomes a smooth $GL_2(\mathbb{A}^\infty)$-module by right translation. But notice that the space $\mathcal{A}$ depends on the choice of $K$ and $GL_2(\mathbb{R})$ does not act on $\mathcal{A}$ by right translation: conjugates of $K$ are not commensurable (usually have trivial intersection), so does not preserve the $K$-finiteness. Nevertheless, the infinite part does acts on $\mathcal{A}$ via the notion of $(\mathfrak{g}_\mathbb{C}, K)$-modules: it is both a $\mathfrak{g}_c$-module and $K$-module and the two actions are compatible on $\mathfrak{k}=\Lie K$.
Definition 15 Let $V$ be a $(\mathfrak{g}_\mathbb{C} ,K)$-module. We say that $V$ is admissible if for any $\sigma: K\rightarrow GL_n(\mathbb{C})$ continuous irreducible, the isotypic subspace $V[\sigma]\subseteq U$ is finite dimensional. We say $V$ is irreducible if it is algebraically irreducible, i.e. there is no proper $(\mathfrak{g}_\mathbb{C},K)$-submodule.
Remark 20 One can check that $\mathcal{A}$ is a $(\mathfrak{g}_\mathbb{C},K)$-module.
Remark 21 For the space $\mathcal{A}_0$ of cusp forms, one can complete it with respect respect to the unitary structure and obtain a genuine action of $GL_2(\mathbb{R})$. For the space $\mathcal{A}$, one can also obtain a genuine action of $GL_2(\mathbb{R})$ but the the completion depends on a choice of the norm. The miracle is that the $(\mathfrak{g}_\mathbb{C},K)$-module structure is "minimal" for the purpose of studying automorphic forms.
Definition 16 A $(\mathfrak{g}_\mathbb{C},K)\times GL_2(\mathbb{A}^\infty)$-module is a $\mathbb{C}$-vector space $V$ endowed with the structure of $(\mathfrak{g}_\mathbb{C},K)$-module and a commuting smooth action of $GL_2(\mathbb{A}^\infty)$. We say $V$ is admissible if for any open compact subgroup $U\subseteq GL_2(\mathbb{A}^\infty)$ and for all continuous irreducible representations $\sigma: U\times K\rightarrow GL_n(\mathbb{C})$, the isotypic subspace $V[\sigma]\subseteq V$ is finite dimensional. Similarly, we say $V$ is irreducible is it is algebraically irreducible.
Proposition 5 Suppose $\pi$ is an irreducible admissible $(\mathfrak{g}_\mathbb{C}, K)\times GL_2(\mathbb{A}^\infty)$-module, then there exists $\pi_\infty$ as an irreducible admissible $(\mathfrak{g}_\mathbb{C} ,K)$-module and for all $p$, an irreducible admissible $GL_2(\mathbb{Q}_p)$-module, unramified for almost all $p$, such that $$\pi\cong \left(\bigotimes_p{}'\pi_p\right)\otimes_\mathbb{C} \pi_\infty.$$
Proof See Flath in Corvallis I.
Theorem 7
  1. The spaces $\mathcal{A}_0$ and $\mathcal{A}$ are naturally $(\mathfrak{g}_\mathbb{C}, K)\times GL_2(\mathbb{A}^\infty)$-modules.
  2. For any $v\in \mathcal{A},\mathcal{A}_0$, $v$ generates an admissible $(\mathfrak{g}_\mathbb{C},K)\times GL_2(\mathbb{A}^\infty)$-module.
  3. $\mathcal{A}_0$ is semisimple (but $\mathcal{A}$ is not).
Proof This is theorem that requires real work. For part b): one needs reduction theory for $GL_2$ and also finiteness results of solutions to certain differential equations (highly nontrivial). For part c), one needs a generalization of Peterson inner product and the "unitary implies semisimple" result.
Remark 22 The admissibility fails without the moderate growth condition. For example, it is also interesting study the functions on the punctured Riemann sphere $Y(1)=SL_2(\mathbb{Z})\backslash \mathcal{H}$ that have arbitrary singularity at the puncture, but the space of such functions are no longer finite dimensional. From the view of Galois representations, it suffices consider to the functions with moderate growth.
Definition 17 An automorphic representation of $GL_2(\mathbb{A})$ is an irreducible $(\mathfrak{g}_\mathbb{C},K)\times GL_2(\mathbb{A}^\infty)$-module which is isomorphic to a subquotient of $\mathcal{A}$. It is cuspidal if it is isomorphic to a subquotient of $\mathcal{A}_0$.
Remark 23 Every automorphic representation is admissible.
Proposition 6
  1. Let $k\ge1$. We define a $(\mathfrak{g}_\mathbb{C}, K)$-module $D_{k-1}$ as follows. As a vector space $$D_{k-1}=\bigoplus_{n\equiv k(2),|n|\ge k}\mathbb{C} e_n.$$ As a $K$-module, $$r_\theta e_n=e^{in \theta}e_n,\quad \left(\begin{smallmatrix}1 & 0\\ 0 &-1\end{smallmatrix}\right) e_n=e_{-n}.$$ As a $\mathfrak{g}_c$-module, $$X_+e_n=(k+n)e_{n+2},\quad X_-e_n=(k-n)e_{n-2},\quad Ze_n=(2-k)e_n$$ and $we_n$ is defined to be the differentiate action of $K^0$. Then $D_{k-1}$ is an irreducible admissible $(\mathfrak{g}_\mathbb{C},K)$-module.
  2. If $f\in C^\infty(GL_2(\mathbb{R}))$ satisfies $X_-f=0$ and $f(\left(\begin{smallmatrix}z &0 \\0 & z\end{smallmatrix}\right) gr_\theta)=e^{ik\theta}z^{2-k}f(g)$, then $f$ generates a $(\mathfrak{g}_\mathbb{C},K)$-submodule of $C^\infty(GL_2(\mathbb{R}))$ which is isomorphic to $D_{k-1}$.
Corollary 4 There are isomorphisms of smooth $GL_2(\mathbb{A}^\infty)$-modules: $$\Hom_{(\mathfrak{g}_\mathbb{C},K)}(D_{k-1},\mathcal{A})\cong \mathcal{M}_k,\quad \Hom_{(\mathfrak{g}_\mathbb{C},K)}(D_{k-1},\mathcal{A}_0)\cong \mathcal{S}_k,$$ given by $f\mapsto f(e_k)$.

In particular, there is a bijection between irreducible $GL_2(\mathbb{A}^\infty)$-submodules of $\mathcal{S}_k$ and irreducible $(\mathfrak{g}_\mathbb{C},K)\times GL_2(\mathbb{A}^\infty)$-submodules $\pi$ of $\mathcal{A}_0$ such that $\pi_\infty\cong D_{k-1}$.

Remark 24 In other words, the classical modular forms are picked out by the $\pi_\infty$ component of the corresponding automorphic representation $\pi$ being the discrete series (and the limit of discrete series when $k=1$) $D_{k-1}$. There are also principal series representation of $GL_2(\mathbb{R})$ depending on continuous parameters, which correspond to the Maass forms. We will discuss them in detail but rather put them in more general framework: we will start to define reductive groups and introduce automorphic representations on general reductive groups. Miraculously we don't need know too much about the reductive groups themselves in order to do so.


TopReductive groups over algebraically closed fields

Convention. We will assume that $F$ is a field of characteristic 0. By a variety over $F$ we mean a reduced scheme of finite type over $F$ (not necessarily connected or irreducible). Any affine scheme is separated so we will not assume separatedness.

Definition 18 An algebraic group $G/F$ is a variety endowed with morphisms $m: G\times G\rightarrow G$, $i: G\rightarrow G$ and a point $e\in G(F)$ making $G$ a group object in the category of varieties. A representation of $G$ is a pair $(\pi, V)$ where $V$ is a finite dimensional $F$-vector space and $\pi:G\rightarrow GL(V)$ is a homomorphism of algebraic groups. We say $G$ is a linear algebraic group if it admits a faithful representation.
Definition 19 A linear algebraic group $G$ is reductive if it is geometrically connected and every representation of $G$ is semisimple.
Example 7 $G=GL_n$ is a reductive group. If $F$ is a number field and $G/F$ is a reductive group, we will discuss a good theory of automorphic forms on $G(F)\backslash G(\mathbb{A}_F)$, generalizing the classical case $F=\mathbb{Q}$ and $G=GL_2$.

Reductive groups are nice from the perspective of representation theory. Even better, one can classify all reductive groups over an algebraically closed field using certain combinatoric data and classify them over general fields via descent.

Convention. In this section we assume that $F$ is algebraically closed.

Definition 20 A torus $T$ is a linear algebraic group such that there exists an isomorphism $T\cong \mathbb{G}_m^n$ for some $n\ge0$. We define the character group and the cocharacter group of $T$, $$X^\cdot(T)=\Hom(T, \mathbb{G}_m),\quad X_\cdot(T)=\Hom(\mathbb{G}_m,T).$$ These are free abelian groups of rank $n $.
Lemma 4
  1. The natural pairing $X^\cdot(T)\times X_\cdot(T)\rightarrow \Hom(\mathbb{G}_m, \mathbb{G}_m)\cong \mathbb{Z}$ is a perfect paring.
  2. The assignment $T\mapsto X^\cdot(T)$ is an equivalence of categories between the category of $F$-tori and the category of finitely generated free abelian groups.
  3. Every irreducible representation of $T$ is 1-dimensional and every representation of $T$ is semisimple (so $T$ is reductive).
Example 8 Any representation of $\mathbb{G}_m$ is of the form $t\mapsto\diag\{t^{a_1},\ldots, t^{a_n}\}$, where $a_i\in \mathbb{Z} $.
Definition 21 The derived group of a linear algebraic group $G$ is defined to be $DG=\cap_N N$, where $N $ runs over all closed normal subgroup such that $G/N$ is abelian. The derived central series is $G\supseteq DG\supseteq DDG\supseteq\cdots$. We say $G$ is solvable if this series terminates at the trivial group.

We recall the Jordan decomposition: If $V$ is a finite dimensional $F$-vector space and $g\in GL(V)(F)$, then there exists unique commuting elements $g_s, g_u\in GL(V)(F)$ such that $g=g_sg_u$,where $g_s$ is diagonalizable and $g_u$ is unipotent. From this one can deduce the following theorem.

Theorem 8 Let $G$ be a linear algebraic group and $g\in G(F)$. Then there unique exists commuting elements $g_s,g_u\in G(F)$ such that $g=g_sg_u$ and for any representation $(\pi, V)$ of $G$, $\pi(g_s)$ is semisimple and $\pi(g_u)$ is unipotent.
Definition 22 A linear algebraic group $G$ is unipotent of every $g\in G(F)$ is unipotent.

The following proposition provides a intuitive way to think about the various notion of linear algebraic groups.

Proposition 7
  1. Every connected solvable group $G$ admits a faithful representation $\pi: G\rightarrow GL(V)$ with image in the subgroup of upper triangular matrices.
  2. Every unipotent group $U$ (in characteristic 0 it is automatically connected) admits a faithful representation $\pi: G\rightarrow GL(V)$ with image in the subgroup of the unipotent upper triangular matrices.
  3. Every connected solvable group $G$ contains a unique normal unipotent group such that $G/U$ is a torus (we haven't defined quotients, one way to think about it is that there exists a torus $T$ such that $$1\rightarrow U\rightarrow G\rightarrow T\rightarrow 1$$ is exact in $F$-points). We call $U$ the unipotent part of $G$.
Definition 23 The radical $RG$ of a linear algebraic group $G$ is defined to be the maximal connected normal solvable subgroup. The unipotent radical $R_uG$ is defined to be the unipotent part of $RG$.
Proposition 8 Let $G$ be a connected linear algebraic group. The $G$ is reductive if and only if $R_uG$ is trivial.
Remark 25 In positive characteristics $p>0$, the reverse direction fails: the only connected groups with all representations semisimple are the tori. For example, let $V$ the the standard representation of $SL_2(F)$, then $V^{\otimes p}=F\{x^ky^{p-k}\}_k$ contains a sub-representation generated by $\{x^p,y^p\}$ and is not semisimple. So we instead use "$R_uG$ is trivial" for the definition of reductive groups in general.

Now we fix a reductive group $G$.

Definition 24 We say a torus $T\subseteq G $ ($\subseteq$ means a closed subgroup) is maximal if it is not contained in any strictly larger torus.
Proposition 9 Let $T\subseteq G $ be a torus. Then
  1. The centralizer $Z_G(T)$ is reductive and connected.
  2. The normalizer has identity component $N_G(T)^0=Z_G(T)$. In particular, the quotient $N_G(T)/Z_G(T)$ is finite.
  3. $T$ is maximal if and only if $Z_G(T)=T$.

The finite group $W(G)=N_G(T)/Z_G(T)$ is called the Weyl group of $(G,T)$.

Fix a maximal torus $T\subseteq G $. Then $G$ has a natural action on $\mathfrak{g}=\Lie G=T_e(G)$: $\Ad: G\rightarrow GL(\mathfrak{g})$ given by differentiating the conjugation. Restricting to $T$ we obtain a natural representation of $T$. This gives the Cartan decomposition $$\mathfrak{g}=\mathfrak{g}_0\oplus \bigoplus_\alpha \mathfrak{g}_\alpha,$$ where $\mathfrak{g}_\chi$ is the $\chi$-eigenspace of $\Ad|_T$ for $\chi\in X^\cdot(T)$.

Proposition 10 $\mathfrak{g}_0=\Lie T$. Each $\mathfrak{g}_\alpha$ is either 0 or 1-dimensional.
Definition 25 The elements $\alpha\in X^\cdot(T)$ such that $\alpha\ne0$ and $\mathfrak{g}_\alpha\ne0$ are called the roots of $(G,T)$. We write $\Phi\subseteq X^\cdot(T)$ for the set of roots.
Definition 26 If $\alpha$ is a root, we define $T_\alpha=\ker(\alpha)^0\subseteq T$ and $G_\alpha=Z_G(T_\alpha)$ (a reductive subgroup of $G$), then $T\subseteq G_\alpha$ is a maximal torus again.
Proposition 11 The Weyl group $W(G_\alpha,T)$ contains a unique nontrivial element $s_\alpha$ (so $W(G_\alpha,T)$ has order 2) and there exits a unique element $\alpha^\vee\in X_\cdot(T)$ such that $s_\alpha(x)=x-\langle\alpha^\vee,x\rangle \alpha$ for all $x\in X^\cdot(T)$. Namely, $s_\alpha$ acts on $X^\cdot(T)$ by reflection and this reflection can be chosen using an integral element $\alpha^\vee\in X_\cdot(T)$.
Definition 27 The element $\alpha^\vee$ is called the coroot of the root $\alpha$. We write $\Phi^\vee\subseteq X_\cdot(T)$ for the set of coroots. We have a canonical bijection between $\Phi$ and $\Phi^\vee$ given by $\alpha\mapsto\alpha^\vee$.

The tuple $(X^\cdot(T), \Phi,X_\cdot(T), \Phi^\vee)$ becomes a root datum in the following sense.

Definition 28 A root datum is a tuple $(M,\Psi,M^\vee,\Psi^\vee)$ with a bijection between $\Phi$ and $\Psi^\vee$ and a perfect pairing $\langle,\rangle: M\times M^\vee\rightarrow \mathbb{Z}$, where $M$ and $M^\vee$ are finitely generated free abelian groups and $\Psi\subseteq M-\{0\}$, $\Psi^\vee\subseteq M^\vee-\{0\}$ are finite subsets such that
  1. For any $\alpha\in \Psi$, $\langle\alpha,\alpha^\vee\rangle=2$.
  2. For any $\alpha\in \Psi$, the automorphism $s_\alpha(x)=x-\langle x, \alpha^\vee\rangle\alpha$ of $M$ leaves $\Psi$ invariant.
  3. The subgroup of $\Aut(M)$ generated by the $s_{\alpha}$'s is finite.
Example 9 The most fundamental case is $G=SL_2$. Choose $T=\{s(t)=\diag\{t,t^{-1}\}\}$. We can easily check that $Z_G(T)=T$ and hence $T$ is a maximal torus. The Lie algebra $\mathfrak{g}$ is spanned by $H=\left(\begin{smallmatrix}1 & 0\\ 0 & -1\end{smallmatrix}\right)\in \Lie T$, $E=\left(\begin{smallmatrix}0 & 1\\ 0 & 0\end{smallmatrix}\right)$ and $F=\left(\begin{smallmatrix}0 & 0 \\ 1& 0\end{smallmatrix}\right)$, as $\Ad((s(t))(E)=t^2E$, $\Ad(s(t))(F)=t^{-2}E$. We obtain the Cartan decomposition $$\mathfrak{g}=\mathfrak{g}_0\oplus \mathfrak{g}_\alpha\oplus \mathfrak{g}_{-\alpha},$$ where $\alpha(s(t))=t^2$ and $\Phi=\{\alpha,-\alpha\}\subseteq X^\cdot(T)\cong \mathbb{Z}$.

Notice $\ker \alpha=\diag\{\pm1,\pm1\}$, so $T_\alpha=1$ and $G_\alpha=G$. A representatives for the nontrivial element in the Weyl group is $w=\left(\begin{smallmatrix}0 & 1\\-1&0\end{smallmatrix}\right)$. One checks that $ws(t)w^{-1}=\diag\{t^{-1}, t\}=s(t)^{-1}$, i.e., $s_\alpha$ acts as $-1$ on $X^\cdot(T)$ and $\alpha^\vee=s\in X_\cdot(T)$. After choosing the natural basis, one finds that the root datum of $SL_2$ is $(\mathbb{Z},\{2,-2\}, \mathbb{Z}, \{1,-1\})$.

Example 10 $G=GL_n$. The torus $T=\diag\{\lambda_1,\ldots,\lambda_n\}$ is a maximal torus. The normalizer $N_G(T)$ is the set of monomial matrices (i.e., with exactly one entry in each row and column). The Weyl group $W(G,T)$ is isomorphic $S_n$. Let $e_i\in X^\cdot(T)$ be the character $\diag\{\lambda_1,\ldots,\lambda_n\}\mapsto\lambda_i$. Then $e_1,\ldots, e_n$ is a basis of $X^\cdot(T)$. The Cartan decomposition is given by $$\mathfrak{g}=\Lie T\bigoplus_{1\le i,j\le n} E_{ij}.$$ Notice that $\Ad(t)E_{ij}=(e_i-e_j)(t)E_{ij}$, So the set of roots $\Phi=\{e_i-e_j: i\ne j\}$.

Suppose $\alpha=e_1-e_2$, then $\ker(\alpha)=\diag\{\lambda,\lambda,\lambda_2,\ldots\lambda_n\}$ and $G_\alpha=\diag\{\left(\begin{smallmatrix}{*} & {*}\\ {*} & {*}\end{smallmatrix}\right), \lambda_3,\ldots,\lambda_n\}\cong GL_2\times \mathbb{G}_m^{n-2}$. So $s_\alpha$ is represented by $\diag\{\left(\begin{smallmatrix}0 &1\\1&0\end{smallmatrix}\right) ,1,\ldots1\}$ and $s_\alpha$ is simply the permutation $(12)\in S_n$. More generally, if $\alpha=e_i-e_j$, then $s_\alpha=(ij)\in S_n$. The coroot associated to $\alpha=e_i-e_j$, then $\alpha^{\vee}(t)=\diag\{1,\ldots, t,\ldots, t^{-1},\ldots1\}$, where $t$ is in the $i$-th entry and $t^{-1}$ is in the $j$-th entry. The root datum of $GL_n$ is then $\{\mathbb{Z}^n,\Phi,\mathbb{Z}^n,\Phi^\vee\}$, where $\Phi=\{e_i-e_j: i\ne j\}$ and $\Phi^\vee=\{\varepsilon_i-\varepsilon_j: i\ne j\}$ and $\{\varepsilon_i\}$ forms a basis for $X_\cdot(T)$.

Theorem 9 Let $G$ be a reductive group.
  1. $G$ contains a unique $G$-conjugacy class of maximal tori (this implies that the root datum of $(G,T)$ is independent of choice of $T$).
  2. (Isomorphism theorem) The root datum of $G$ determines $G$ up to isomorphism.
  3. (Existence theorem) Every abstract root datum is equivalent to the root datum of a reductive group.


Definition 29 Let $G$ be a reductive group. A Borel subgroup $B\subseteq G$ is a maximal connected solvable subgroup (not normal in general). A parabolic subgroup $P\subseteq G $ is a subgroup such that $G/P$ is projective as an algebraic variety.
Theorem 10 Let $G$ be a reductive group.
  1. $G$ contains a unique conjugacy class of Borel subgroups.
  2. A subgroup $P\subseteq G $ is parabolic if and only if it contains a Borel subgroup. In particular, Borel subgroups are parabolic.
Example 11 $G=GL_n$. Fix a partition $n=n_1+\cdots +n_r$ into positive (= nonnegative) integers . Let $P\subseteq G $ be the block upper triangular subgroup with blocks of size $n_1,\ldots,n_r$. Then $G/P$ is the Grassmanian of filtrations $0\subseteq \Fil^1\subseteq \cdots\subseteq \Fil^r=F^n$ such that $\dim_F \Fil^i/\Fil^{i-1}=n_i$, which is a projective variety. For the partition $n=1+1+\cdots+1$, we obtain the subgroup of upper triangular matrices, a Borel subgroup of $G$.
Definition 30 Fix $T\subseteq G $ a maximal torus. Let $(M,\Phi,M^\vee,\Phi^\vee)$ be the root datum of $(G,T)$. A root basis of $(M,\Phi,M^\vee,\Phi^\vee)$ is a subset $S\subseteq \Phi$ such that every $\beta\in \Phi$ can uniquely expressed as $\beta=\Sigma_{\alpha\in S}n_\alpha \alpha$, where $n_\alpha$ are integers that are all positive or all negative.
Remark 26
  1. Root bases always exist.
  2. Having fixed a root basis $S$, we get a decomposition $\Phi=\Phi^+\cup \Phi^-$ into the set of positive roots and the set of negative roots. We call $S$ the set of simple roots.
Proposition 12 There is a canonical bijection between the root bases and the Borel subgroups $B $ that contains $T$, characterized by $\Lie B=\Lie T\bigoplus_{\alpha\in \Phi^+}\mathfrak{g}_\alpha$ inside the Cartan decomposition of $\mathfrak{g}$. The Weyl group acts simply transitively on both sides of this bijection.
Example 12 $G=GL_n$. The Borel subgroups $B\supseteq T$ is in bijection with the orderings $\sigma\in S_n$ of the standard basis $\{v_1,\ldots,v_n\}$ of $F^n$, i.e., $B_\sigma$ is given by the stabilizer of the flag $\{\Fil^i=\langle v_{\sigma(1)},\ldots v_{\sigma(i)}\rangle\}_{i=1}^n$. For example, $\sigma=\Id$ gives the subgroup of upper triangular matrices $B $. The corresponding root basis is $\{e_i-e_{i+1}\}_{i=1}^{n-1}$ and $\Lie B=\Lie T\bigoplus_{i<j} E_{ij}=\Lie T\bigoplus_{i<j}\mathfrak{g}_{e_i-e_j}$ as desired.
Definition 31 A based root datum is a tuple $(M,\Phi,S,M^\vee,\Phi^\vee,S^\vee)$ where $S$ is a root basis of the root datum $(M,\Phi,M^\vee,\Phi^\vee)$. So a choice of $T\subseteq B\subseteq G$ gives a based root datum.
Remark 27 If $P\subsetneq G$ is a parabolic subgroup, then it is not reductive but there exists an exact sequence $$1\rightarrow R_uP\rightarrow P\rightarrow L\rightarrow1,$$ where $L $ is a reductive group. In fact, the sequence is always split (the splitting depends on the choice of a maximal torus $T$). So there exists a closed reductive subgroup $L'\subseteq P$ such that the maps $R_uP\times L'\rightarrow P$ and $L'\rightarrow L$ are isomorphisms. This $L'$ is unique up to conjugation by $R_uP$. We call any such $L'$ a Levi subgroup and the decomposition $P=R_uP\times L'$ a Levi decomposition.
Definition 32 A reductive group $G$ is semisimple if $RG$ is trivial. A semisimple group $G$ is almost-simple if it has no normal subgroup of dimension $>0$; simple if it has no nontrivial normal subgroup.
Remark 28 If $G$ is a reductive group, then $RG=Z(G)^0$, a torus. The derived subgroup $DG$ is semisimple. Moreover, the map $DG\times RG\rightarrow G$ is surjective with finite kernel. So classifying reductive groups reduces to classifying semisimple groups and finite central subgroups: every reductive group is quotient by a product of semisimple groups and tori by finite groups.
Definition 33 An isogeny of semisimple groups is a surjective homomorphism $f:G\rightarrow G'$ with finite (automatically central) kernel. If $G$ is semisimple, we say $G$ is simply-connected if there is no proper isogeny $f: G'\rightarrow G$; adjoint if there is no proper isogeny $f: G\rightarrow G'$ (equivalently, $G$ has trivial center).
Remark 29 If $G$ is a semisimple group, then there exist normal pairwise commuting almost simple subgroups $G_1,\ldots G_r\subseteq G$ such that the homomorphism $G_1\times \cdots G_r\rightarrow G$ is surjective with finite kernel. Unlike the case of elliptic curves, the isogeny relation is not symmetric. The equivalence relation generated by the isogeny relation gives isogeny classes.
Remark 30 For any reductive group $G$, we have $\mathbb{Z}\Phi\subseteq M\subseteq (\mathbb{Z}\Phi^\vee)^*\subseteq M \otimes \mathbb{Q}$. One can show that $G$ is semisimple if and only if $\mathbb{Z} \Phi\subseteq $ is of finite index.

Let $f: G'\rightarrow G$ be an isogeny. Let $T'{}=f^{-1}(T)^0\subseteq G'$. This is a maximal torus of $G'$ and $f|_{T'}:T'\rightarrow T$ is an isogeny of tori. It induces $f^*|_{T'}: X^\cdot(T')\rightarrow X^\cdot (T)$, hence an isomorphism $X^\cdot(T')\otimes \mathbb{Q}\cong X^\cdot(T) \otimes \mathbb{Q}$. We define $N=f^*|_{T'}^{-1}(X^\cdot(T))\subseteq M \otimes \mathbb{Q}$. Then $(N,\Phi, N^\vee,\Phi^\vee)$ is also a root datum, thus $\mathbb{Z} \Phi\subseteq N\subseteq (\mathbb{Z}\Phi^\vee)^*$. In other words an isogeny $f:G'\rightarrow G$ gives a subgroup $N/\mathbb{Z}\Phi$ of the finite group $(\mathbb{Z} \Phi^\vee)^*/(\mathbb{Z}\Phi)$. This is true for any groups in the isogeny class of $G$.

Theorem 11 If $G$ is a semisimple group, then there is an ordering-preserving bijection between the elements of the isogeny class of $G$ and the subgroups of $(\mathbb{Z} \Phi^\vee)^*/(\mathbb{Z}\Phi)$. In particular, the simply-connected group corresponds to the full group and the adjoint group corresponds to the trivial group.
Definition 34 If $G$ is almost simple, we associate to it a graph, the Dynkin diagram of $G$. Fix a root basis $S\subseteq \Phi$. The Dynkin diagram has the vertex set $S$. For $\alpha,\beta\in S$, let $n(\beta,\alpha)=\langle\beta,\alpha^\vee\rangle\in \mathbb{Z}$. Then $n(\beta,\alpha)n(\alpha,\beta)\in \{0,1,2,3\}$ and we join $\alpha$,$\beta$ with $n(\beta,\alpha)n(\alpha,\beta)$ edges. If $n(\beta,\alpha)n(\alpha,\beta)\ge2$, we decorate this multiple edge with an arrow from $\beta$ to $\alpha$ if $|n(\beta,\alpha)|>|n(\alpha,\beta)|$.
Theorem 12 Let $G$ be an almost simple group. The Dynkin diagram of $G$ is connected. If $G'$ is another almost simple group, then $G$ and $G'$ are in the same isogeny class if and only if they have the same Dynkin diagram.
Definition 35 Let $G$ be a reductive group. We denote by $\Aut(G)$ the group of automorphisms of $G$ as an algebraic group over $F$. We have an exact sequence $$1\rightarrow\Inn(G)\rightarrow\Aut(G)\rightarrow \Out(G)\rightarrow1.$$ Notice $\Inn(G)=G/Z(G)$.
Definition 36 A pinning of $G$ is a tuple $(T, B, \{x_\alpha\}_{\alpha\in S})$ where $T$ is a maximal torus, $B\supseteq T$ a Borel subgroup, $S\subseteq \Phi(G,T)$ a root basis of $B $ and $X_\alpha$ is a basis of $\mathfrak{g}_\alpha$.
Definition 37 Let $\mathcal{P}$ be a pinning and $R$ be a based root datum of $\mathcal{P}$. We define $\Aut(G,\mathcal{P})\subseteq \Aut(G)$ be the automorphisms of $G$ that fixes $T,B$ and the set $\{X_\alpha\}$. We define $\Aut(R)$ be the automorphisms of $M$ that fixes $\Phi$ and $S$. There are natural maps $\Aut(G,\mathcal{P})\rightarrow\Out(G)$ and $\Aut(G,\mathcal{P})\rightarrow\Aut(R)$.
Theorem 13
  1. The natural maps $\Aut(G,\mathcal{P})\rightarrow\Out(G)$ and $\Aut(G,\mathcal{P})\rightarrow\Aut(R)$ are isomorphisms. Consequently, $\Aut(G)\cong\Inn(G)\rtimes\Aut(G,\mathcal{P})\cong\Inn(G)\rtimes\Aut(R)$.
  2. All pinning of $G$ are $\Inn(G)$-conjugate (so all the above splitting are conjugate under the action of $\Inn(G)$).

TopReductive groups over general fields

In this section we assume that $F$ is an arbitrary field of characteristic 0. Let $G$ be an algebraic group over $F$. Let $\bar F$ be an algebraic closure of $F$.

Definition 38 We say $G$ is reductive, semisimple, unipotent, solvable, torus if $G/\bar F$ satisfies the corresponding property.

Notice $\Gamma_F=\Gal(\bar F/F)$ acts on $G(\bar F)$ with fixed points $G(F)$.

Definition 39 A torus $T/F$ is called split if there exists an isomorphism $T\cong\mathbb{G}_m^n$ defined over $F$ (such an isomorphism always exists over $\bar F$). If $T$ is any torus over $F$. We define $X^\cdot(T)=\Hom(T/\bar F,\mathbb{G}_m/\bar F)$. This is a free finitely generated abelian group with an action of the Galois group $\Gamma_F$.
Theorem 14 The assignment $T\mapsto X^\cdot(T)$ defines an (anti-)equivalence of categories between the category of tori over $F$ and the category of free finitely generated abelian groups with a smooth $\Gamma_F$-action (equivalently, the conjugacy classes of homomorphisms $\Gamma_F\rightarrow GL_n(\mathbb{Z})$.)


Remark 31 Let $\Gamma_F\rightarrow GL_n(\mathbb{Z})$ be the homomorphisms associated to $T/F$. Then $T(F)\cong(\bar F^\times\otimes_\mathbb{Z} \Hom(\mathbb{Z}^n,\mathbb{Z}))^{\Gamma_F}$, where $\Gamma_F$ acts diagonally. For example, when $n=1$, the homomorphism $\Gamma_F\rightarrow GL_1(\mathbb{Z})\cong \{\pm\}$ corresponds to quadratic extensions $K/F$ or the trivial extension $F$. When $K/F$ is a quadratic extension, then $T(F)=\{\lambda\in K^\times: \bar \lambda=\lambda^{-1}\}=\{\lambda\in K^\times: \mathbb{N}_{K/F}(\lambda)=1\}$. In particular, for the quadratic extension $\mathbb{C}/\mathbb{R}$, we obtain that $T(\mathbb{R})=S^1$.
Definition 40 We say that a torus $T\subseteq G $ is maximal if $T_{\bar F}\subseteq G_{\bar F}$ is maximal. It is a fact that $G$ always contains maximal tori.
Definition 41 A maximal split torus $T\subseteq G $ is a split torus, maximal with respect this property. We way $G$ is split if $G$ contains a split maximal torus.
Remark 32 Notice that a split maximal torus is not the same as a maximal split torus. For example, there exist groups $G$ that contains no nontrivial split torus, so the latter exists but the former doesn't. It is a fact that there exists a unique $G(F)$-conjugacy class of maximal split tori.

We start off with the split reductive groups when we classify reductive groups over general fields.

Theorem 15
  1. Let $R$ be a root datum. Then there exists a reductive group $G/F$ and a split maximal torus $T\subseteq G $ such that $R$ is the root datum of $(G,T)$.
  2. If $G$ and $G'$ be split reductive groups over $F$ with the same root datum, then they are isomorphic over $F$.
Definition 42 Let $G$ be a linear algebraic group over $F$. A form of $G$ is a linear algebraic group $G'$ over $F$ such that there exists an isomorphism $G_{\bar F}\cong G'{}_{\bar F}$. By the above theorem, any reductive group over $F$ is a form of a split reductive group over $F$.

So our remaining task is to classify the forms of a split group.

Let $G/F$ be a split reductive group and $G'/F$ be any reductive group with $f:G_{\bar F}\cong G'{}_{\bar F}$. For any $\sigma\in \Gamma_F$, $\sigma$ acts on $f$ and obtain another ${}^\sigma f: G_{\bar F}\cong G'{}_{\bar F}$. One knows that $f^{-1}\circ {}^\sigma f\in\Aut(G_{\bar F})$. This is an example of a 1-cocycle in non-abelian Galois cohomology.

Definition 43 Let $A $ be a group with a continuous action of $\Gamma_F$ ($A $ is endowed with the discrete topology). A 1-cocycle is a map $\Gamma_F\rightarrow A,\quad a \mapsto a_\sigma$ satisfying $a_{\sigma\tau}=a_\sigma\cdot{}^\sigma a_\tau$. We say two 1-cocycles $a_\sigma,b_\sigma$ are 1-cohomologous if there exists $c\in A$ such that $b_\sigma=c^{-1}a_\sigma{}^\sigma c$ for all $\sigma\in \Gamma_F$.
Remark 33
  1. $H^1(F,A)$ is a pointed set: the the identity class is given by $a_\alpha=\Id_A$.
  2. For any linear algebraic group $H $, $H^1(F,H(\bar F))$ and $H^1(F,\Aut(H_{\bar F}))$ are well defined pointed sets.
  3. Any $\Gamma_F$-equivariant homomorphism $A\rightarrow B$ induces a natural map $H^1(F,A)\rightarrow H^2(F,B)$.
  4. If $\Gamma_F$ acts trivially on $A $, then $H^1(F,A)$ is the set of homomorphisms $\Gamma_F\rightarrow A$ up $A $-conjugacy.

We have constructed a 1-cocycle $a_\sigma=f^{-1}\circ{}^\sigma f$. Changing $f$ to $f\circ h$ where $h\in \Aut(G_{\bar F})$ doesn't change the class of $a_\sigma$. Therefore we obtain a map between the isomorphism classes of forms of $G$ to $H^1(F,\Aut(G_{\bar F}))$. Using Galois descent one can show that

Proposition 13 This map is bijective.

Suppose $G$ is split and fix a pinning $\mathcal{P}=(T,B,\{X_\alpha\}_{\alpha\in S})$ defined over $F$ (i.e., $T\subseteq G $ is split, $B\subseteq G$ is defined over $F$ and $X_\alpha\in \mathfrak{g}_\alpha$ are defined over $F$). Let $R$ be the associated based root datum. Recall (Theorem 13)the pinning $\mathcal{P}$ splits the exact sequence $$0\rightarrow\Inn(G_{\bar F})\rightarrow\Aut(G_{\bar F})\rightarrow \Out(G_{\bar F})\cong \Aut(R)\rightarrow0.$$ $\Gamma_F$ acts on this exact sequence and one can show that when $\mathcal{P}$ is defined over $F$, $\Gamma_F$ acts trivially on $\Out(G_{\bar F})\cong\Aut(R)$ and the splitting is $\Gamma_F$-equivariant. Thus it gives a homomorphism $$H^1(F, \Aut(G_{\bar F}))\rightarrow H^1(F,\Out(G_{\bar F}))=\{\text{homo. }\mu: \Gamma_F\rightarrow\Aut(R)\}/\text{conj.}$$ In particular, any form $G'$ of $G$ gives a homomorphism $\mu_G: \Gamma_F\rightarrow\Aut(R)$ up to conjugacy (which of course does not determine $G/F$ up to isomorphisms because we lose information when passing from $\Aut$ to $\Out$.)

Definition 44 A reductive group over $F$ is quasi-split if it contains a Borel subgroup (i.e. $B\subseteq G$ such that $B_{\bar F}\subseteq G_{\bar F}$ is a Borel subgroup).
Remark 34
  1. Split groups are quasi-split (one can easily write down a Borel subgroup over $F$ using the root datum).
  2. Being quasi-split is very restrictive.
Definition 45 Let $G,G'$ be reductive groups over $F$. We say $G'$ is an inner form of $G$ if there exists an isomorphism $f: G_{\bar F}\cong G'_{\bar F}$ such that for any $\sigma\in \Gamma_F$, $f^{-1}\circ{}^\sigma f\in\Inn(G_{\bar F})$. Equivalently, the associated class $c_{G,G'}\in H^1(F,\Aut(G_{\bar F}))$ lies in the image of $H^1(F,\Inn(G_{\bar F}))\rightarrow H^1(F,\Aut(G_{\bar F}))$.
Remark 35 The relation of inner forms is symmetric.

The following lemma tells us that passing to $\Out$, we exactly lose the information distinguishing the inner forms of the same quasi-split group.

Lemma 5 $G$ has a unique quasi-split inner form. If $G'$ and $G''$ are forms of the same split group $G$ , then they have the same quasi-split inner form if and only $\mu_{G'}\cong\mu_{G''}$ up to conjugacy.
Remark 36 Given a split group $G$ and a $F$-pinning $\mathcal{P}$, a homomorphism $\mu:\Gamma_F\rightarrow\Aut(R)$ gives an element of $H^1(F,\Aut(G_{\bar F}))$ using the splitting provided by the $F$-pinning. The corresponding form $G_\mu$ of $G$ is then a quasi-split group.
Remark 37 Two groups having the same quasi-split inner forms are closely related to each other (e.g., they have the same Langlands dual group, which controls the information about automorphic forms on them).

We summarize the strategy of classifying reductive groups over general fields:

  1. Construct the split group $G$ given a root datum.
  2. Classify the quasi-split groups (corresponding to homomorphisms $\Gamma_F\rightarrow\Out(G)$).
  3. Classify the inner forms of each quasi-split group.

The example $G=GL_n$ is in order.

Example 13 If $A $ is a central simple algebra of rank $n $, we define a group $G_A$ over $F$ by the functor of points: $G_A(R)=(R \otimes_F A)^\times$ (this is a representable functor). $G_A$ depends on $A $ only up to isomorphism, so $$G_{A,\bar F}\cong G_{A_{\bar F}}=G_{M_n(\bar F)}=GL_{n,\bar F}.$$ Every inner form of $G=GL_n$ is isomorphic to $G_A$ for some $A $. For example, when $F=\mathbb{R}$, the group $GL_2$ has two inner forms $GL_2(\mathbb{R})$ and $\mathbb{H}^\times$.

Fix the standard pinning $\mathcal{P}$ of $GL_n$. The associated based root datum $R$ has automorphism group $\Aut(R)\cong \mathbb{Z}/ 2 \mathbb{Z}$. So there is a bijection between quasi-split (non-split) forms of $GL_n$ and quadratic extensions $K/F$. Define a Hermitian form $\langle\cdot,\cdot\rangle$ on $K^n$ by $\langle x,y\rangle={}^t\bar x J y$, where $J=\mathrm{antidiag}\{1,-1,1,\ldots,(-1)^{n-1}\}\in M_n(F)$. Notice $g\mapsto {}^tg^{-1}$ does map to the nontrivial element in $\Aut(R)$, but the choice of $J$ is chosen to be compatible with the splitting given by the standard pinning: $g\mapsto J^{-1}{}^tg^{-1}J$ preserves the standard Borel $B $ and the alternating signs are chosen to preserve each basis $X_\alpha$.

Define the group $G^K$ by the functor of points $$\{G^K(R)=g\in GL_2(R \otimes _F K): {}^t \bar g J g=J\}.$$ The group $G^K$ becomes isomorphic to $GL_n$ over $K$. Indeed, for any $K$-algebra $R$, we have $K \otimes_F R\cong R\times R$ and the automorphism $x \otimes y\mapsto x \otimes \bar y$ of $K \otimes_F R$ induces the automorphism $(x,y)\mapsto (y,x)$ of $R\times R$. It follows that $G^K(R)\cong GL_n(R)$. One can show the upper triangular subgroup of $G^K$ is a Borel subgroup, therefore $G^K$ is quasi-split as desired.

The inner forms of $G^K$ can be constructed as follows: let $B $ be a central simple algebra over $K$ and ${*}$ be an anti-involution of $B $ that restricts to $x\mapsto \bar x$ on $K$ (e.g., $B=M_n(K)$, $g^*= \bar J^{-1}{}^t \bar g J$). Then each inner form of $G^K$ is of the form $$G_{(B,*)}(R)=\{g\in (B \otimes_K R)^\times: g^*g=1\}.$$


TopAutomorphic representations on reductive groups

Reference on this section: the article of Borel-Jacquet in Corvallis I.

Definition 46 Let $F$ be a number field and $G/F$ be a reductive group. Choose an embedding $G\hookrightarrow GL_{n,F}$, then for any place $v$ of $F$, we endow $G(F_v)$ the subspace topology from $GL_{n,F_v}$. For $v$ a finite place, we define $K_v=G(F_v)\cap GL_n(\mathcal{O}_{F_v})$ an open compact subgroup of $G(F_v)$. If we choose another embedding $G\hookrightarrow GL_{n',F}$, then $K_v=K_v'$ for almost all $v$. We define $G(\mathbb{A}_F)$ to be the restricted product $\prod'G(F_v)$ with respect to $K_v$. The topology is the subspace topology from $GL_n(\mathbb{A}_F)$.
Remark 38 One can always obtain an integral model of $G$ over $\mathcal{O}_F$ by taking the Zariski closure in $GL_{n,\mathcal{O}_F}$ and $K_v$ is the $\mathcal{O}_{F_v}$-points of that integral model. But usually the model thus obtained is very singular at some places. Throwing away the bad places defines a good integral model of $G$ over a certain the ring of $S$-integers.
Definition 47 Recall that for any Hausdorff locally compact topological group $H $, there is a Haar measure on $H $ (Definition 6) unique up to $\mathbb{R}_{>0}$-multiple. The modulus character $\delta_H: H\rightarrow \mathbb{R}_{>0}$ is defined as $$\delta_{H}(g)\int_H f(hg)dh=\int_H f(h)dh.$$ $H $ is called unimodular if $\delta_H$ is trivial.
Remark 39 For any reductive group or unipotent group $G$ over a number field $F$, $G(F_v)$ (for any $v$) and $G(\mathbb{A}_F)$ are all unimodular. Indeed one can construct the Haar measure using algebraic differential top forms on $G$. The right translation acts on the 1-dimensional spaces of such top forms and induces a character of $G$. But there are no nontrivial characters of a semisimple or unipotent group. Solvable groups are not unimodular in general and Borel subgroups are never unimodular.
Definition 48 If we further assume $Z\subseteq H$ is a closed unimodular subgroup. Then there exists a Haar integral $\mu: C_c^\infty(Z\backslash H)\rightarrow \mathbb{C}$ on the quotient space $Z\backslash H$ which is right $H $-invariant. Again $\mu$ is unique up to $\mathbb{R}_{>0}$-multiple.

Denote $G_\infty=G(F \otimes_\mathbb{Q} \mathbb{R})$, a real Lie group. We choose $K_\infty\subseteq G_\infty$ a maximal compact subgroup (which is unique up to conjugacy). Let $\mathfrak{g}_0=\Lie G$ and $\mathfrak{g}=\mathfrak{g}_0 \otimes_\mathbb{R} \mathbb{C}$.

Definition 49 An automorphic form on $G$ is a function $\phi: G(\mathbb{A}_F)\rightarrow \mathbb{C}$ satisfying:
  1. For any $\gamma\in G(F)$, $g\in G(\mathbb{A}_F)$, $\phi(\gamma g)=\phi(g)$.
  2. For any $g^\infty\in G(\mathbb{A}_F^\infty)$, the function $g_\infty\mapsto\phi(g^\infty g_\infty)$ is smooth.
  3. $\phi$ is $Z(\mathfrak{g})$-finite, i.e., $Z(\mathfrak{g})\cdot \phi$ is a finite dimensional vector space.
  4. $\phi$ is $K_\infty$-finite, i.e., $\mathbb{C}[K_\infty]\cdot \phi$ is a finite dimensional vector space.
  5. There exists $K^\infty\subseteq G(\mathbb{A}_F^\infty)$ an open compact subgroup such that for any $g\in G(\mathbb{A}_F)$ and $k^\infty\in K^\infty$, $\phi(gk^\infty)=\phi(g)$.
  6. For any $g^\infty\in G(\mathbb{A}_F^\infty)$, the function $g_\infty\mapsto \phi(g^\infty g_\infty)$ is slowly increasing. A function $f: G_\infty\rightarrow \mathbb{C}$ is slowly increasing, if for one (equivalently, all) embeddings $\pi: G_\infty\hookrightarrow GL_{n,\mathbb{R}}$ of real algebraic groups, there exists $c, N>0$ such that for any $g\in G_\infty$, $$|f(g)|\le c|\tr {}^t\pi(g)\pi(g)+\tr {}^t\pi(g^{-1})\pi(g^{-1})|^N.$$

We say $\phi$ is cuspidal if it further satisfies

  1. For all parabolic subgroups $P\subseteq G $ defined over $F$, for any $g\in G(\mathbb{A}_F)$, $$\int_{N(F)\backslash N(\mathbb{A}_F)} f(ng)dn=0,$$ where $N $ is the unipotent radical of $P$. Notice by the left-$G(F)$ invariance, if this condition is satisfied for a parabolic subgroup $P $, then it is satisfied for all parabolic subgroups conjugate to $P $.

We write $\mathcal{A}$ for the space of automorphic forms on $G$ and $\mathcal{A}_0\subseteq \mathcal{A}$ the subspace of cusp forms.

Remark 40 Notice the quotient $N(F)\backslash N(\mathbb{A}_F)$ is always compact (it reduces to the fact that $F\backslash\mathbb{A}$ is compact). One recovers the definition of cuspidality for $GL_2$ using the standard Borel subgroup of upper triangular matrices.
Remark 41 Over an algebraically closed field, the parabolic subgroups (containing a fixed maximal torus) are in bijection with the subsets of nodes of the Dynkin diagram. Over a general field $F$, the $F$-parabolic subgroups (containing a fixed minimal parabolic subgroup) are similarly classified using the relative root system. There are always finitely many $G(F)$-conjugacy classes in a $G(\bar F)$-conjugacy class of parabolic subgroups and there is a unique $G(F)$-conjugacy class of minimal parabolic subgroups.
Remark 42 As the case for $GL_2$, the spaces $\mathcal{A}$, $\mathcal{A}_0$ are $(\mathfrak{g},K_\infty)\times G(\mathbb{A}_F^\infty)$-modules.
Remark 43 By a theorem of Harish-Chandra, if $f$ is $Z(\mathfrak{g})$-finite and $K_\infty$-finite, then $f$ is analytic. If $f$ is further slowly increasing, then $f$ has uniform moderate growth: for any $D\in U(\mathfrak{g})$, $Df$ is slowly increasing. Indeed, there exists $\alpha\in C_c^\infty(G_\infty)$ such that $f*\alpha=f$, therefore $Df=D(f*\alpha)=f*D\alpha$. Functions of the form $u*\alpha$ are all slowly increasing.
Theorem 16 (Harish-Chandra) If $\phi\in\mathcal{A}$, then it generates an admissible $(\mathfrak{g},K_\infty)\times G(\mathbb{A}_F^\infty)$-submodule of $\mathcal{A}$.
Definition 50 An automorphic representation $\pi$ on $G$ is an indecomposable $(\mathfrak{g},K_\infty)\times G(\mathbb{A}_F^\infty)$-module, isomorphic to a subquotient of $\mathcal{A}$. We say $\pi$ is cuspidal if it is isomorphic to a subquotient of $\mathcal{A}_0$ ($\mathcal{A}_0$ is semisimple as the case $G=GL_2$).
Remark 44
  1. An automorphic representation is always admissible (by the previous theorem).
  2. An automorphic representation always has a factorization $\pi\cong \pi_\infty \otimes \otimes_{v\nmid \infty}' \pi_v$ just as the case of $GL_2$. More on this later.
  3. If $F$ is an arbitrary field and $G$ a reductive group over $F$, we say $G$ is anisotropic if very maximal split torus is trivial. If $G$ is $F$-anisotropic, then it contains no nontrivial parabolic subgroups over $F$. If $F$ is a number field and $G$ is $F$-anisotropic, then $\mathcal{A}=\mathcal{A}_0$ since the cuspidality condition is vacuous.
Remark 45 One should think about the cusps using the quotient $G(F)\backslash G(\mathbb{A}_F)$. This quotient is compact if and only if $G$ is $F$-anisotropic. When the quotient is non-compact, the cusps exactly correspond to the conjugacy classes of $F$-parabolic subgroups. One can also reformulate of cuspidality condition by the "rapidly decreasing at $\infty$" condition.
Example 14 (Hilbert modular forms) Let $F/\mathbb{Q}$ be a totally real field. Let $v_1,\ldots, v_d$ be the infinite places. Let $G=GL_{2,F}$. Then $G(F \otimes_\mathbb{Q} \mathbb{R})=\prod_{i=1}^d GL_2(\mathbb{R})$. We choose $K_\infty=\prod_{i=1}^d O_2(\mathbb{R})$. Recall for any $k\ge1$ we defined an irreducible, admissible $(\mathfrak{gl}_2, O_2(\mathbb{R}))$-module $D_{k-1}$ (Proposition 6). We choose integers $k_1,\ldots, k_d\ge1$ and consider the cuspidal automorphic representations $\pi$ of $G$ such that $\pi_\infty D_{k_1-1}\otimes \cdots \otimes D_{k_d-1}$. These correspond to classical Hilbert modular forms of weight $(k_1,\ldots, k_d)$.
Example 15 (Quaternion algebras over $\mathbb{Q}$) Let $A $ be quaternion algebra over $\mathbb{Q}$. Let $G=G_A$ be the associated inner form of $GL_{2,\mathbb{Q}}$. Recall that for $F=\mathbb{Q}_p$ or $\mathbb{R}$, there exists two isomorphic classes of quaternion algebras over $F$, the split one and the non-split one. For $\mathbb{Q}$, the isomorphic classes of quaternion algebras correspond to finite set of places $S$ of $\mathbb{Q}$ of even cardinality, where $S$ is the set of non-split places of $A $.

Suppose $S$ is nonempty and does not contain $\infty$, then $G_{A_S}$ is anisotropic modulo center since $G_{\mathbb{Q}_p}$ is anisotropic for every $p\in S$. Moreover, $G(\mathbb{R})\cong GL_2(\mathbb{R})$. One can check that cuspidal automorphic representations of $G(\mathbb{A})$ such that $\pi_\infty\cong D_{k-1}$ can be described in terms of holomorphic functions on a compact quotient of $\mathcal{H}$: there are no cusps since $G_{A_S}$ is anisotropic.

Now suppose $S$ contains $\infty$. Then $G$ is still anisotropic modulo center but $G(\mathbb{R})\cong\mathbb{H}^\times$ is compact modulo center. The automorphic representations $\pi$ such that $\pi_\infty$ is trivial are in bijection with the $GL_2(\mathbb{A}^\infty)$-constituents of the space $\phi: G(\mathbb{A})\rightarrow \mathbb{C}$ such that

  1. For any $\gamma\in G(\mathbb{Q})$, $\phi(\gamma g)=\phi(g)$.
  2. There exists $K^\infty\in G(\mathbb{A}^\infty)$ an open compact subgroup which fixes $\phi$ on the right.
  3. For any $g_\infty\in G(\mathbb{R})$, $g\in G(\mathbb{A})$, $\phi(g g_\infty)=\phi(g)$ (due to the triviality at $\infty$).

In other words, these are the functions on a finite set $\phi: G(\mathbb{Q})\backslash G(\mathbb{A}^\infty)/K^\infty\rightarrow \mathbb{C}$. It turns out these simple-looking functions are very interesting and the following Jacquet-Langlands correspondence is the first case of functoriality:

Theorem 17 (Jacquet-Langlands) The following sets are in bijection.
  1. Infinite dimensional automorphic representations of $G(\mathbb{A})$ such that $\pi_\infty$ is trivial.
  2. Cuspidal automorphic representations of $GL_2(\mathbb{A})$ such that $\pi_\infty\cong D_1$ and for $p\in S\backslash\infty$, $\pi_p$ is square-integrable.
Remark 46 Let $E/\mathbb{Q}$ be an elliptic curve and $\pi$ be the associated cuspidal automorphic representation of $GL_2(\mathbb{A})$. Then $\pi_p$ is square-integrable if $E $ has multiplicative reduction at $p$. The function on the finite set given by the Jacquet-Langlands correspondence indeed encodes tons of arithmetic information of $E $.


TopRepresentation theory of $p$-adic groups

Reference for this section:

  • Bushnell, Henniart, Local Landlands conjecture for GL(2);
  • Cartier, Corvallis I.

To talk about automorphic representations more intelligently, we need to know more representation theory of local fields, both non-archimedean and archimedean. Then Dick Gross will come and tell you about $L $-groups. Afterwards we are going to discuss Galois representations and it relation with automorphic representations, and as promised, the theorem of Chenevier-Clozel on number fields with limited ramification.

TopUnramified representations

Let $F$ be a finite extension of $\mathbb{Q}_p$ and $G/F$ be an reductive group. We abuse notion and write $G$ for the locally compact topological group $G(F)$. Today we will focus on the simplest case: the unramified representations of a unramified $p$-adic reductive group.

Definition 51 $G$ is called unramified if $G$ is quasi-split and split over an unramified extension of $F$. In other words, $G$ is determined by a homomorphism $\mu_G: \Gamma_F\rightarrow\Aut(R)$ that factors through an unramified extension of $F$.

The nice thing about unramified groups is that they admit distinguished open compact subgroups.

Proposition 14 Let $G^\mathrm{ad}=G/Z(G)$. Then $G=G(F)$ contains a unique $G^\mathrm{ad}(F)$-conjugacy class of open compact subgroup $K$, called the hyperspeicial maximal compact subgroup, characterized by one of the following conditions:
  1. There exists a smooth affine group scheme over $\mathcal{G}$ the ring of integers $A $ such that $\mathcal{G}_F=G$ and the special fiber $\mathcal{G}_{A/(\pi)}$ is a connected algebraic group with trivial unipotent radical and $K=\mathcal{G}(A)$.
  2. If $\mu$ is a Haar measure on $G$ and $K'\subseteq G$ another open compact subgroup, then $\mu(K)\ge\mu(K')$.
Example 16 $G=GL_n(F)$ has a hyperspeicial maximal compact subgroup $K=GL_2(\mathcal{O}_F)$. Because any compact subgroup of $G$ can be conjugated into $GL_2(\mathcal{O}_F)$, we easily see that there exists a unique conjugacy class of such hyperspeicials.

$G=SL_n(F)$ has a hyperspeicial maximal compact subgroup $SL_n(\mathcal{O}_F)$. There are $n $ $G$-conjugacy classes of hyperspeicial maximal compact subgroups which form a single $PGL_n(F)$-conjugacy class. Notice $PGL_n(F)/\Im(SL_n(F))\cong F^\times/(F^\times)^n$, so the $G$-conjugacy class obtained using $g\in PGL_n(F)$ is determined by the valuation of determinant of $g$ mod $n $.

Definition 52 Let $G$ be unramified and fix $S\subseteq G $ a maximal split torus. Let $T=Z_G(S)$, then $T$ is a maximal torus (for general $G$, it is the "anisotropic kernel" of $G$). We denote by $T^c\subseteq T$ the maximal compact subgroup.
Example 17 Let $G=GL_n(F)$ and $S$ be the diagonal torus. Then $T$ is also the diagonal torus and $T^c=A^\times\times\cdots A^\times$.
Definition 53 The split Weyl group $W_d=N_G(S)/Z_G(S)$ is a constant group scheme over $F$ It acts faithfully on $S$ and $T$.
Theorem 18 (Satake isomorphism) There exists a canonical isomorphism between the Hecke algebras $\mathcal{H}(G,K)\cong\mathcal{H}(T,T^c)^{W_d}$, where $K$ is a hyperspeicial maximal compact subgroup of $G$.
Proof We can assume $T^c=T\cap K$ after conjugation. We will write down the Satake transform $\mathcal{S}:\mathcal{H}(G,K)\rightarrow\mathcal{H}(T,T^c)$. Let $B\subseteq G$ be a Borel subgroup containing $T$ (because $G$ is quasi-split, $G$ contains a Borel; a maximal torus is the centralizer of the its maximal split subtorus and all maximal split tori are conjugate, we can conjugate such a Borel to contain $T$ ). Let $\delta_B: B\rightarrow \mathbb{R}_{>0}$ be the modulus character. Given $F\in \mathcal{H}(G,K)$, we define $$\mathcal{S}(f)(t)=\delta_{B}^{1/2}(t)\int_{n\in N}f(tn)dn,$$ where $N $ is the unipotent radical of $B $. One can check that it has image in $\mathcal{H}(T,T^c)^{W_d}$. It turns out the map does not depend on the choice of $B $ (due to the modulus character factor). For details, see Cartier.
Definition 54 Let $\pi$ be an irreducible admissible representation of $G$. We say $\pi$ is $K$-unramified if $\pi^K\ne0$. We say a character $\chi: T\rightarrow \mathbb{C}^\times$ unramified if it is trivial on $T^c$.
Remark 47 A warning: it is necessary to specify $K$ in the previous definition. If $G=SL_2(F)$ and $K,K'$ be non-conjugate hyperspeicial maximal compact subgroup, one can find a representation $\pi$ such that $\pi^K\ne0$ and $\pi^{K'}=0$. This is related the phenomenon of $L $-packets and does not occur for $GL_n$.

The previous theorem together with Theorem 3 have the following corollaries.

Corollary 5 If $\pi$ is $K$-unramified, then $\dim\pi^K=1$.
Corollary 6 The following sets are in canonical bijections:
  1. The set of isomorphism classes of $K$-unramified representations $\pi$.
  2. The set of $W_d$-conjugacy classes of unramified characters $\chi: T\rightarrow \mathbb{C}^\times$.
  3. The set of $W_d$-conjugacy classes of homomorphisms $X_\cdot(S)\rightarrow \mathbb{C}^\times$.
  4. The set of algebra homomorphisms $\mathbb{C}[X_\cdot(S)]^{W_d}\rightarrow \mathbb{C}^\times$.
Proof The equivalences of the first two and last two are clear. For the equivalence between b) and c), simply notice that $X_\cdot(S)\rightarrow T/T^c, \lambda\mapsto\lambda(\pi)$ is a group isomorphism.

Now we introduce the notion of parabolic induction for general reductive groups $G$ and come back to unramified representations in a moment.

Definition 55 Let $P\subseteq G $ be a parabolic subgroup and $P=MN$ be the Levi decomposition. Let $(\sigma,W)$ be an admissible representation of $M$. We define a $G$-representation $$\Ind_P^G\sigma=\{f: G\rightarrow W\text{ locally constant}: f(pg)=\sigma(p)f(g)\},$$ where $G$ acts by right translation.
Proposition 15 $\Ind_P^G\sigma$ is admissible.
Proof Consider any open compact subgroup $K\subseteq G$. We need to show that $(\Ind_P^G\sigma)^K$ is finite dimensional. Since $G/P$ is projective and hence compact, we can find $X\subseteq G $ a finite set such that $PXK=G$. Then $f\in (\Ind_P^G\sigma)^K$ is determined by the values $f(x)$ for $x\in X$. One can check that $f(x)\in W^{xKx^{-1}\cap P}$. The latter space is finite dimensional as $\sigma$ is admissible by assumption.
Definition 56 If $(\sigma,W)$ is an admissible representation of $M$. We define the normalized or unitary induction $\iota_P^G\sigma=\Ind_P^G\sigma \otimes \delta_P^{1/2}$.
Remark 48 The modulus character of a parabolic is equal to $\delta_P(p)=|\det\Ad_\mathfrak{n}(p)|$, where $\mathfrak{n}=\Lie N$.
Proposition 16 Suppose $\sigma$ is unitary. Then $\iota_P^G\sigma$ is also unitary.
Proof We sketch the case $\sigma: M\rightarrow \mathbb{C}^\times$ is a character. Then $\sigma$ is unitary means that $\sigma\bar \sigma=1$. Then $$\iota_P^G\sigma=\{f: G\rightarrow \mathbb{C}: f(pg)=\delta_P^{1/2}(p)\sigma(p)f(pg)\}.$$ For $f_1,f_2\in \iota_P^G$, we write $F(g)=f_1(g)\bar f_2(g)$ and try to define the unitary structure by integrating $F$. However, $P$ is not unimodular and there does not exist $G$-invaraint integrals on the quotient $P\backslash G$. Nevertheless, we do have a right $G$-invariant integral $f\mapsto \int_{P\backslash G}f(x)dx$ on the subspace of locally constant functions $f:G\rightarrow \mathbb{C}$ such that $f(pg)=\delta_P(p)f(g)$. One can check that $F(pg)=\delta_P(p)F(g)$ using the extra $\delta_P$ factor and $\sigma\bar\sigma=1$. So integrating $F$ does give a unitary structure on $\iota_P^G\sigma$.

Now we come back the the situation where $G$ is unramified, $K\subseteq $ hyperspeicial, $S\subseteq G$ maximal split torus. $T=Z_G(S)$ and $B\subseteq G$ a Borel containing $T$. Let $\chi: T\rightarrow \mathbb{C}^\times$ be any smooth character.

Definition 57 Define $I(\chi)=\iota_B^G\chi$. This is called a principal series representation of $G$. If $\chi$ is unramified, then $I(\chi)^K$ is 1-dimensional. As a consequence, $I(\chi)$ has exactly one $K$-unramified subquotient, denoted by $\pi_\chi$. Conversely, if $\pi$ is $K$-unramified, then we get (up to $W_d$-conjugacy) an unramified character $\chi: T\rightarrow \mathbb{C}^\times$ via the Satake isomorphism and $\pi\cong\pi_\chi$.
Remark 49 For any smooth character $\chi: T\rightarrow \mathbb{C}^\times$ and $w\in W_d$, $\iota_B^G(\chi)$ and $\iota_B^G(\chi^w)$ are not necessarily isomorphic but they are both of finite length with the same Jordan-Holder factors. Indeed one can represent the character of the both representations using induced data (certain orbital integrals) and compute the action of $w$ on these integrals directly.
Example 18 $G=GL_n$, $S=T$ is the diagonal torus, $K=GL_n(\mathcal{O}_F)$, $B $ is the standard Borel. Then $W_d=S_n$ and $\mathcal{H}(G,K)\cong \mathbb{C}[e_1^\pm,\ldots,e_n^\pm]^{S_n}\cong \mathbb{C}[T_1,\ldots, T_n,T_n^{-1}]$, where $e_i$ is the standard cocharacter $t\mapsto \diag\{1,\ldots,1,t,1,\ldots1\}$ and $T_i$ is the $i$-th symmetric polynomial in $e_1,\ldots, e_n$.
Example 19 Let $f\in S_k(\Gamma_0(N))$ be a classical holomorphic modular forms that is an $T_p$-eigenvector for all $p\nmid N$ with eigenvalue $a_p$. $f$ generates an automorphic representation $\pi$ of $GL_2(\mathbb{A})$. For $p\nmid N$, $\pi_p$ is $K_p$-unramified for $K_p=GL_2(\mathbb{Z}_p)$. Factor the Hecke polynomial $$x^2-a_px+p^{k-1}=(x-p^{1/2}\alpha_1)(x-p^{1/2}\alpha_2).$$ Then the character associated to $\pi_p$ via the Satake isomorphism is $$\chi: T(\mathbb{Q}_p)\rightarrow \mathbb{C}^\times: \diag\{p^a, p^b\}\mapsto\alpha_1^a\alpha_2^b.$$ One can compute directly that $I(\chi)$ is irreducible and $\pi_p\cong I(\chi)$. Explicitly, $$I(\chi)=\{f: GL_2(\mathbb{Q}_p)\rightarrow \mathbb{C}: f(\left(\begin{smallmatrix}a & b\\ 0 & d\end{smallmatrix}\right)g)=\alpha_1^{\ord_p(a)}\alpha_2^{\ord_p(d)}|a/d|^{1/2}f(g) \}.$$


Lemma 6 Let $F$ be a number field and $G$ be a reductive group over $F$. For all but finitely many finite places $v$ of $F$. The group $G_{F_v}$ is unramified and the group $K_v=G(F_v)\cap GL_n(\mathcal{O}_{F_v})$ is a hyperspeicial maximal compact subgroup.
Proof (Sketch) There is an $\Aut(G)$-torsor $\Iso(G, G^\mathrm{qs})$, where $G^\mathrm{qs}$ is the quasi-split inner form of $G$. This torsor has a marked connected component $\mathrm{InnIso}(G, G^\mathrm{qs})$ that is a torsor for $G^\mathrm{ad}$. The assertion that $G_{F_v}$ is quasi-split almost everywhere is equivalent to the assertion that $\mathrm{InnIso}(G,G^\mathrm{qs})(F_v)$ is non-empty for almost every $v$.
Remark 50 If $U\subseteq G(\mathbb{A}_F^\infty)$ is an open compact subgroup. Then there exists $V\subseteq U$ an open compact of the form $V=\prod_i V_v$ with $V_v=K_v$ hyperspeicial for almost all $v$. If $\pi$ is an irreducible admissible representation of $G(\mathbb{A}_F^\infty)$, then there exists such a $V$ such that $\pi^V\ne0$.
Proposition 17 If $\pi$ is an irreducible admissible representation of $G(\mathbb{A}_F^\infty)$, then for all $v\nmid \infty$, there exists an irreducible admissible representation $\pi_v$ of $G(F_v)$ and an isomorphism $\pi\cong\bigotimes_v{}'\pi_v$. In particular, $\pi_v^{K_v}\ne0$ for almost all $v$ (hence is 1-dimensional for almost all $v$).
Proof See Flath, Corvallis I.

TopHierarchy of representations of $p$-adic groups

Let us come back to the local situation. Let $F$ be a $p$-adic field and $G/F$ be a reductive group. We are going to define several classes of irreducible admissible representations of $G(F)$.

Lemma 7 Let $(\pi,V)$ be an irreducible admissible representation of $G(F)$. Then there is a central character $\omega: Z(F)\rightarrow \mathbb{C}^\times$ such that $\pi(z)(v)=\omega(z)v$ for any $z\in Z(F)$ and $v\in V$.
Proof See Bushnell-Henniart.
Definition 58 We define $\pi^\vee$, the contragredient of $\pi$ as the space of smooth vectors in the algebraic dual $\Hom_\mathbb{C}(V, \mathbb{C})$ (this definition works for arbitrary smooth representations). If $\pi$ is admissible (not necessarily irreducible), then the natural map $\pi\rightarrow(\pi^\vee)^\vee$ is an isomorphism.
Definition 59 Let $(\pi,V)$ be an irreducible admissible representations of $G(F)$ with central character $\omega$. Choose $v\in V$, $v^\vee\in V^\vee$. We define the matrix coefficient $c_{v,v^\vee}(g)=\langle\pi(g)v,v^\vee\rangle$, where $\langle,\rangle$ is the natural pairing between $V$ and $V^\vee$.
Definition 60 We say $\pi$ is square-integrable if $\omega$ is unitary (i.e. $\omega\bar\omega=1$) and for any $v\in V$, $v^\vee\in V^\vee$, $$\int_{Z(F)\backslash G(F)}|c_{v,v^\vee}(g)|^2dg<\infty.$$
Remark 51 The integral is well-defined because both $Z(F)$ and $G(F)$ are unimodular and the function $g\mapsto |c_{v,v^\vee}(g)|^2$ descent s to $Z(F)\backslash G(F)$ since $\omega$ is unitary.
Definition 61 We say $\pi$ is supercuspidal if for any $v\in V$, $v^\vee\in V^\vee$, the function $g\mapsto c_{v,v^\vee}(g)$ is compactly supported modulo center.
Proposition 18
  1. If $\pi$ is irreducible admissible, then $\pi$ is square-integrable (resp., supercuspidal) if and only if there exists nonzero $v,v^\vee$, such that $c_{v,v^\vee}$ is square-integrable modulo center (resp. compactly supported modulo center) and $\omega$ is unitary (resp. no condition on $\omega$).
  2. If $\omega$ is unitary, then $\pi$ is supercuspidal implies that $\pi$ is square-integrable.
  3. If $\pi$ is square-integrable, then $\pi$ is unitary (i.e., $V$ admits a $G$-invariant positive definite inner product).
Proof (Sketch)
  1. Fix $v^\vee$, let $W\subseteq V$ be the space of $v\in V$ such that $c_{v,v^\vee}$ is compactly supported modulo center. Then one can check that $W\subseteq V$ is a nonzero $G$-invariant subspace, hence $W=V$.
  2. Obvious.
  3. Fix $v^\vee\in V^v$ nonzero. We define for $v,w\in V$, $$(v,w)=\int_{Z(F)\backslash G(F)}c_{v,v^\vee}(g)\overline{c_{w,v^\vee}(g)}dg.$$ One can check it defines a unitary structure.
Definition 62 Let $\pi$ be irreducible admissible with $\omega$ unitary. We say $\pi$ is tempered if for any $v,v^\vee$, $c_{v,v^\vee}$ lies in $L^{2+\varepsilon}(Z(F)\backslash G(F))$ for any $\varepsilon>0$.
Remark 52 We have the following hierarchy of irreducible admissible representations with unitary central character: then all representations $\supseteq$ unitary $\supseteq$ tempered $\supseteq$ square-integrable $\supseteq$ supercuspidal.

So far everything works well for all locally profinite groups. The following theorem needs more from input from the structure theory of reductive groups.

Theorem 19
  1. An irreducible admissible representation $(\pi, V)$ of $G(F)$ is supercuspidal if and only if there does not exist a proper parabolic $P=MN\subsetneq G$ and an admissible representation $\sigma$ of $M$ together with an embedding $\pi\hookrightarrow \iota_P^G\sigma$.
  2. Any irreducible admissible representation admits an embedding $\pi\hookrightarrow \iota_P^G\sigma$ for some $P=MN\subseteq G$ and $\sigma$ a supercuspidal representation of $M$.
Proof This is a hard theorem: see Casselman unpublished notes on $p$-adic groups.
Remark 53 Since $G/P$ is compact, $\iota_P^G\sigma$ is always of finite length. Decomposing $\iota_P^G\sigma$ is not completely trivial. One can always arrange that $\pi$ is the unique irreducible quotient for some $\sigma$ (Langlands quotient).

TopLocal Langlands correspondence for $GL_n$ over $p$-adic fields

Example 20 $G=GL_n$. A standard parabolic $P\subseteq G $ corresponds to a partition $n=n_1+\cdots+n_r$, where $n_i\ge1$. Its Levi subgroup $M=GL_{n_1}\times \cdots \times GL_{n_r}$. Therefore every irreducible admissible representation $\pi$ of $GL_n(F)$ embeds as $\pi\hookrightarrow \iota_P^G(\sigma_1 \otimes \cdots \otimes\sigma_r)$, where $\sigma_i$ is a supercuspidal for $GL_{n_i}(F)$. A special case we have seen is that any unramified representation $\pi$ is a subquotient of $\iota_B^G(\chi_1 \otimes \cdots \otimes\chi_n)$, where $\chi_i: GL_1(F)\rightarrow \mathbb{C}^\times$ is an unramified character.

There is a nice interpretation in terms the local Langlands correspondence for $GL_n(F)$ proved by Harris-Taylor and Henniart. The local Langlands correspondence for $GL_n(F)$ is a bijection $\mathrm{rec}_F$ between

  1. isomorphism classes of irreducible admissible representations of $GL_n(F)$;
  2. conjugacy classes of semisimple homomorphisms (i.e. decomposes as direct sums) $\phi: W_F\times SL_2(\mathbb{C})\rightarrow GL_n(\mathbb{C})$ such that $\phi|_{SL_2(\mathbb{C})}$ is algebraic, where $W_F$ is the Weil group of $F$.

$\mathrm{rec}_F$ is characterized abstractly by some identities relating to their $L $-functions.

Example 21 If $\pi$ is the unramified subquotient $\iota_B^G(\chi_1 \otimes \cdots \otimes\chi_n)$, then $\phi=\mathrm{rec}_F(\pi)$ is trivial on $SL_2(\mathbb{C})$, unramified on $W_F$ and sends a uniformizer $\varpi$ to $\diag\{\chi_1(\varpi),\ldots,\chi_n(\varpi)\}$.
Example 22 The local Langlands correspondence $\mathrm{rec}_F$ restricts to a bijection between
  1. classes of supercuspidal representations.
  2. classes of irreducible representations $\phi: W_F\rightarrow GL_n(\mathbb{C})$ (trivial on the $SL_2(\mathbb{C})$ factor).
Example 23 Suppose $n=n_1+\cdots n_r$, and $\phi_i: W_F\rightarrow GL_{n_i}(\mathbb{C})$. Let $\pi_i=\mathrm{rec}^{-1}_F(\phi_i)$, it is a supercuspidal representation of $GL_{n_i}(F)$. Let $\phi=\phi_1\oplus\cdots\phi_r: W_F\rightarrow GL_n(\mathbb{C})$, then $\pi=\mathrm{rec}^{-1}(\phi)$ is indeed a subquotient of $\iota_P^G\pi_1 \otimes\cdots \otimes \pi_r$.

To summarize: building irreducible admissible representations of $GL_n(F)$ from supercuspidal representations of Levi subgroups mirrors taking direct sums of irreducible Langlands parameters $\phi: W_F\rightarrow GL_n(\mathbb{C})$.

Remark 54 More generally, the local Langlands correspondence restricts to a bijection between discrete series representations and irreducible representations $W\times SL_2(\mathbb{C})\rightarrow GL_n(\mathbb{C})$; between tempered representations and representations $W\times SL_2(\mathbb{C})\rightarrow GL_n(\mathbb{C})$ satisfying the weight-monodromy conjecture.
Example 24 Let $n=ab$, $\phi:W_F\rightarrow GL_a(\mathbb{C})$ and $\psi: SL_2(\mathbb{C}) \rightarrow GL_b(\mathbb{C})$ be irreducibles. Then $\mathrm{rec}_F^{-1}(\phi \otimes \psi)$ is a subquotient of $\iota_P^G \sigma|\cdot|^{(b-1)/2} \otimes \cdots \otimes \sigma|\cdot|^{(1-b)/2}$. One can view the $SL_2(\mathbb{C})$-factor a way of labeling different subquotients.

Let $G$ be any reductive group. There is a canonical square-integrable representation of $G$, the Steinberg representation that we are going to construct now. Let $P_0\subseteq G$ be a minimal parabolic, define $$\pi_0=\iota_{P_0}^G\delta_{P_0}^{-1/2}=\Ind_{P_0}^G\mathbf{1}=C_c^\infty(G(F)/P_0(F)).$$

Proposition 19
  1. The subrepresentations $\pi\subseteq \pi_0$ are in bijections with parabolic subgroups $P_0\subseteq P\subseteq G$: $\pi_P=C_c^\infty(G(F)/P(F))$.
  2. Let $\mathrm{St}$ be the quotient of $\pi$ by the span of $\pi_P$, $P_0\subsetneq P$. Then $\mathrm{St}$ is irreducible and square-integrable.
Example 25 If $G=GL_n(F)$. Then $\mathrm{rec}(\mathrm{St})$ is the unique homomorphism trivial on $W_F$ and restricts to the unique $n $-dimensional representation of $SL_2(\mathbb{C})$. Thus $\mathrm{St}$ is a subquotient of $\iota_B^G |\cdot|^{(n-1)/2} \otimes \cdots \otimes |\cdot|^{(1-n)/2}=\iota_B^G\delta_B^{1/2}$. If $G=GL_2(\mathbb{Q}_p)$. Then we have an exact sequence $$1\rightarrow \mathbb{C}\rightarrow C_c^\times(\mathbb{P}^1(\mathbb{Q}_p))\rightarrow1.$$ This exact sequence does not split and gives an example of unramified principal series representation (as $\delta_B$ is unramified) that has a ramified unitary quotient and a trivial subrepresentation. One also knows that $C_c^\infty(\mathbb{P}^1(\mathbb{Q}_p))$ is non unitary since any unitary admissible representation is semisimple (Lemma 3).
Example 26 Let $\pi$ be the cuspidal automorphic representation of $GL_2(\mathbb{A})$ associated to an elliptic curve $E/\mathbb{Q}$. Then $E $ has multiplicative reduction at $p$ if and only if $\pi_p$ is an unramified twist of $\mathrm{St}$.
Remark 55 Constructing supercuspidal representations are highly nontrivial in general. In view of the Langlands parameters: the inertia subgroup of $W_F$ is very complicated.
Example 27 Very recently Gross-Reeder constructed a class of simple supercuspidal representations. For example, consider $F=\mathbb{Q}_2$, $G=SL_2(\mathbb{Q}_2)$ and $I=\{\left(\begin{smallmatrix}a & b \\c &d\end{smallmatrix}\right)\in SL_2(\mathbb{Z}_2): c\in 2 \mathbb{Z}_2\}$. Let $\chi: I\rightarrow \left(\begin{smallmatrix}a & b\\2c& d\end{smallmatrix}\right) \mathbb{C}^\times(-1)^{b+c} $. It is a character and indeed $\pi=c\Ind_I^G\chi$ is a supercuspidal representation.


TopRepresentation theory of real reductive groups

Reference for this section: Wallach in Corvallis I.

Let $G$ be a reductive group $\mathbb{R}$. We use the usual notation: $\mathfrak{g}_0=\Lie G$, $\mathfrak{g}=\mathfrak{g}_0 \otimes_\mathbb{R}\mathbb{C}$, $K\subseteq G(\mathbb{R})$ a maximal compact subgroup. $k_0=\Lie K$, $k=k_0 \otimes_\mathbb{R}\mathbb{C}$.


Definition 63 A representation $(\pi, H)$ of $G(\mathbb{R})$ is a (separable) Hilbert space $H $ and a homomorphism $\pi: G(\mathbb{R})\rightarrow GL(H)$ to the group of bounded invertible linear operators such that
  1. then map $G(\mathbb{R})\times H\rightarrow H$ is continuous.
  2. for any $k\in K$, $\pi(k)$ is unitary (i.e. an isometry).
Remark 56 Given a map satisfying part a) one can always define a new Hilbert space structure on $H $ that also satisfies part b) by the standard averaging argument.
Definition 64 We say $\pi$ is irreducible if there is no nontrivial closed $G(\mathbb{R})$-invariant subspace of $\pi$. We say $\pi$ is unitary if for any $g\in G(\mathbb{R})$, $\pi(g)$ is unitary.

Let us recall the representation theory when $G(\mathbb{R})$ is compact

Theorem 20 (Peter-Weyl) When $G(\mathbb{R})$ is compact.
  1. Any irreducible representation of $G(\mathbb{R})$ is finite dimensional.
  2. Any unitary representation $(\pi, H)$ of $G(\mathbb{R})$ decomposes as a Hilbert direct sum $H\cong\hat\bigoplus_{i\in I} H_i$ of a countable set of irreducible subrepresentations $H_i\subseteq H$.

When $G(\mathbb{R})$ is compact, there is a unique $G(\mathbb{R})$-conjugacy class of maximal tori $T\subseteq G $ (so $T(\mathbb{R})$ is several copies of $S^1$). Fix $T$ a maximal torus and $R$ be an associated root datum. Then the natural map $$N_{G(\mathbb{R})}T(\mathbb{R})/T(\mathbb{R})\rightarrow W(G_\mathbb{C}, T_\mathbb{C})$$ is a bijection. Let $(\pi, H)$ be an irreducible representation of $G(\mathbb{R})$. Then $\pi|_{T(\mathbb{R})}=\bigoplus_\lambda H_\lambda$ as a sum of weight spaces. The map $X^\cdot(T_\mathbb{C})\rightarrow\Hom(T(\mathbb{R}),\mathbb{C}^\times)$ is an isomorphism, so we can view the sum as over $\lambda \in X^\cdot(T_\mathbb{C})$. In particular, $\{\lambda\in X^\cdot(T_\mathbb{C}): H_\lambda\ne0\}$ is invariant under $N_{G(\mathbb{R})}T(\mathbb{R})$, hence under $W(G_\mathbb{C},T_\mathbb{C})$.

Theorem 21 Let $(\pi, H)$ be an irreducible representation of $G(\mathbb{R})$. Fix a root basis $S\subseteq \Phi$.
  1. There exists a unique $\lambda=\lambda_H\in X^\cdot(T_\mathbb{C})$ such that $H_\lambda\ne0$ and for any $\mu\in X^\cdot(T_\mathbb{C})$ with $H_\mu\ne0$, we can write $\mu=\lambda-\sum_{\alpha\in S}n_\alpha\alpha$, where $n_\alpha\in \mathbb{Z}_{\ge0}$.
  2. The assignment $(\pi,H)\mapsto\lambda_H$ defines a bijection between the set of isomorphism classes of irreducible representations and the set of dominant weights $\lambda$ (for any $\alpha\in S$, $\langle\lambda, \alpha^\vee\rangle\ge0$).
Remark 57 If $G$ is any reductive group over $\mathbb{R}$. We say $(\pi_1,H_1)$ and $(\pi_2,H_2)$ are equivalent if there exists a bijective continuous homomorphism $T: H_1\rightarrow H_2$ that intertwines $\pi_1$ and $\pi_2$. If $\pi_1,\pi_2$ are unitary, we say they are unitarily equivalent if $T$ can be chosen to be unitary.
Example 28 If $G=U(n)$ and $T\subseteq G $ be the diagonal maximal torus. So $T(\mathbb{R})=\{e^{i\theta_1},\ldots,e^{i\theta_n}\}$, $G_\mathbb{C}=GL_{n,\mathbb{C}}$ and $T_\mathbb{C}$ is the usual diagonal torus. The theorem then parametrizes the equivalence classes of irreducible representations of $G(\mathbb{R})=U(n)$ by tuples $\lambda_1\ge\cdots\ge\lambda_n$, $\lambda_i\in \mathbb{Z}$.

Top$(\mathfrak{g}, K)$-modules

Let $G$ be any reductive group over $\mathbb{R}$. There is a unique $G(\mathbb{R})$-conjugacy class of maximal compact subgroup $K\subseteq G(\mathbb{R})$ (notice this is not true over $p$-adics, cf., Example 16).

Example 29 $G=SL_n$, $K=SO_n$.
Example 30 $G=U(p,q)$, $K=U(p)\times U(q)$.
Definition 65 A $(\mathfrak{g},K)$-module is a $\mathfrak{g}$-module and $K$-representation $V$ such that
  1. For any $v\in V$, $v$ is $K$-finite, i.e. $W_v=\mathbb{C}[K]v$ is finite dimensional. The map $K\rightarrow GL(W_v)$ is continuous.
  2. For any $X\in k$, $Xv=\frac{d}{dt}(\exp(tX)v)|_{t=0}$.
  3. For any $X\in \mathfrak{g}$ and $k\in K$, $kXv=\Ad(k)(X)kv$.

We say $V$ is admissible if for any irreducible representation $\tau$ of $K$, $\Hom_K(\tau,V)$ is finite dimensional. We say $V$ is unitary if there is a positive definite Hermitian inner product $\langle,\rangle$ such that:

  1. For any $X\in \mathfrak{g}$, $\langle Xv,w\rangle+\langle v, Xw\rangle=0$.
  2. For any $k\in K$, $\langle kv,kw\rangle=\langle v,w\rangle$.
Definition 66 Let $(\pi,H)$ be a representation of $G(\mathbb{R})$. By the Peter-Weyl theorem, $$\pi|_K=\hat\bigoplus_{i\in I}\pi_i,$$ where $\pi_i\subseteq \pi$ are irreducibles. We say $\pi$ is admissible if each isomorphism classes of irreducibles representations of $K$ appears only finitely many times. It is easy to see that if $\pi$ is admissible, then the subspace $H^{K\text{-fin}}\subseteq H$ of $K$-finite vectors is simply the usual direct sum $$H^{K\text{-fin}}\cong\bigoplus_{i\in I}\pi_i.$$ For $X\in \mathfrak{g}$, we define $X_v=\frac{d}{dt}(\exp(tX)v)|_{t=0}$. If this exists, then we say $v$ is differentiable. We say $v$ is smooth if for any $r\ge1$ and any $X_1,\ldots,X_r\in \mathfrak{g}$, $X_1(\cdots(X_rv)\cdots)$ exists.
Theorem 22 Let $(\pi,H)$ be an admissible representation of $G(\mathbb{R})$. Then
  1. Every $K$-finite vector is smooth. $V=H^{K\text{-fin}}$ becomes an admissible $(\mathfrak{g},K)$-module. Moreover, $H $ is irreducible if and only if $V$ is algebraically irreducible as a $(\mathfrak{g},K)$-module.
  2. Every irreducible admissible $(\mathfrak{g}, K)$-module arises in this way from an (but not unique) irreducible admissible representation of $G(\mathbb{R})$.

TopHierarchy of representations of real reductive groups

Definition 67 Let $(\pi_1,H_1)$, $(\pi_2,H_2)$ be two admissible representations of $G(\mathbb{R})$. We say they infinitesimally equivalent if there associated $(\mathfrak{g},K)$ modules are algebraically equivalent.
Remark 58 There do exist infinitesimally equivalent representations which are not equivalent. For example, the notion of parabolic induction depends on a choice of the completion on the space of smooth functions satisfying the usual transformation properties but the different completion turns out to be infinitesimally equivalent. The difference is interesting for certain purposes, nevertheless it does not play a role in the Langlands program: when one restricts to unitary representations, this difference disappears by the following theorem.
Theorem 23 Let $V$ be an admissible $(\mathfrak{g}, K)$-module.
  1. $V$ is unitary if and only if there exists a unitary admissible representation $(\pi,H)$ and $V\cong H^{K\text{-fin}}$ of $(\mathfrak{g},K)$-modules.
  2. If $(\pi_1,H_1)$, $(\pi_2,H_2)$ are irreducible admissible unitary representations. Then they are unitarily equivalent if and only if there is an algebraic isomorphism between their $(\mathfrak{g},K)$-modules preserving the unitary structures.
Definition 68 Let $(\pi,H)$ is an admissible representation of $G(\mathbb{R})$. A matrix coefficient of $\pi$ is a function of the form $g\mapsto c_{v,w}(g)=\langle\pi(g)v,w\rangle$, for $v,w\in H$. It is a $K$-finite matrix coefficient, if $v$ and $w$ are further $K$-finite vectors.
Remark 59 There is a version of Schur's lemma for irreducible admissible representations as the $p$-adic case.
Definition 69 Let $\pi$ be an irreducible representations of $G(\mathbb{R})$ with unitary central character. We say $\pi$ is square-integrable if its $K$-finite matrix coefficients are square-integrable modulo $Z(\mathbb{R})$, where $Z\subseteq G$ is the center as an algebraic group. It is tempered if its $K$-finite matrix coefficients are in $L^{2+\varepsilon}(Z(\mathbb{R})\backslash G(\mathbb{R}))$ for any $\varepsilon>0$. The square-integrable representations are also called discrete series representations since they occur discretely in the unitary dual.
Remark 60 Every tempered or square-integrable representation of $G(\mathbb{R})$ is infinitesimally equivalent to a unitary one. Also notice that $K$-finite matrix coefficients of $\pi$ only depends on the infinitesimal equivalent classes of $\pi$, for $\pi$ irreducible admissible. Therefore we have the following hierarchy of the infinitesimal equivalent classes of irreducible admissible representations with unitary central character: all representations $\supseteq$ unitary $\supseteq$ tempered $\supseteq$ square-integrable.
Remark 61
  1. It follows from Harish-Chandra's work that there is no supercuspidal representations for real reductive groups (in contrast to the $p$-adic case). In view of the Langlands parameters: the Weil group of $p$-adic (especially the wild inertia) is more complicated than the Weil group of $\mathbb{R}$, which is simply a (non-split) extension of $\{1,j\}$ by $\mathbb{C}^\times$.
  2. The irreducible admissible representations of real reductive groups has been completely classified in terms of concrete datum. The key step is to construct discrete series and everything else is relatively easy (in contrast to the $p$-adic case, where the supercuspidal representations are far from being classified).
  3. The local Langlands correspondence is known for any real reductive groups (c.f., the book Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups by Borel-Wallach).

TopThe Harish-Chandra isomorphism

The center $Z(\mathfrak{g})$ of the universal enveloping algebra $U(\mathfrak{g})$ is isomorphic to a polynomial algebra in $r $ generators, where $r $ is the rank of $G_\mathbb{C}$ (= the dimension of a maximal torus). How does one prove it?

We change notion for convenience: let $G$ be a reductive group over $\mathbb{C}$. Fix a pinning $\mathcal{P}$ of $G$.

Definition 70 Let $\rho=\frac{1}{2}\sum_{\alpha\in\Phi^+}\alpha\in X^\cdot(T)\otimes \mathbb{Q}$. Then $\langle\rho,\alpha^\vee\rangle=1$ for any simple root $\alpha\in S$. We have $d\rho\in \Hom(\Lie T, \mathbb{C})=\mathfrak{t}^*$.
Proposition 20 Let $\mathcal{H}=U(\mathfrak{t})\subseteq U(\mathfrak{g})$ (= $\Sym \mathfrak{t}$ because $\mathfrak{t}$ is abelian) and $\mathcal{J}=\sum_{\alpha\in S}U(\mathfrak{g})X_\alpha$. Then $\mathcal{H}$ is a subalgebra and $\mathcal{J}$ is a left ideal of $U(\mathfrak{g})$. We have
  1. $\mathcal{H}\cap \mathcal{J}=0$, $\mathcal{H}\oplus\mathcal{J}\supseteq Z(\mathfrak{g})$.
  2. There exists a unique algebra automorphism $\sigma_S:\mathcal{H}\rightarrow \mathcal{H}$ such that for any $H\in \mathfrak{t} \subseteq\mathcal{H}$, we have $\sigma_S(H)=H-d\rho(H)\cdot1$.
Definition 71 We define $p_S: Z(\mathfrak{g})\rightarrow\mathcal{H}$ to be the projection along $\mathcal{J}$. The Harish-Chandra homomorphism $\gamma: Z(\mathfrak{g})\rightarrow\mathcal{H}$ is defined to be the composite $\gamma=\sigma_S\circ p_S$.
Theorem 24 (Harish-Chandra) The map $\gamma$ is an algebra homomorphism, independent of $S$, and defines a canonical isomorphism $Z(\mathfrak{g})\cong\mathcal{H}^W$, where $W=W(G,T)$ acts on $\mathcal{H}$ in the natural way.


Remark 62 Since $W$ is a reflection group, a nice theorem of Chevalley implies that $\mathcal{H}^W$ is a polynomial algebra in $r $ generators, where $r $ is the rank of $G$.
Example 31 Consider $G=SL_2$ and $\mathfrak{g}=\mathfrak{sl}_2$. We define the Casmir element $\Delta=H^2/2+EF+FE$, where $H=\left(\begin{smallmatrix} 1 & 0 \\ 0 & -1\end{smallmatrix}\right)$, $E=\left(\begin{smallmatrix}0 & 1\\0 & 0\end{smallmatrix}\right)$,$F=\left(\begin{smallmatrix}0 & 0 \\ 1& 0\end{smallmatrix}\right) $. One can directly check that $\Delta\in Z(\mathfrak{g})$. We claim that $Z(\mathfrak{g})=\mathbb{C}[\Delta]$. Since $EF-FE=H$ in $U(\mathfrak{g})$, we can rewrite $\Delta=H^2/2+H+2FE$. Fix the root basis $\{\alpha\}$, where $\alpha(\diag\{t,t^{-1}\})=t^2$. Then $E $ is a basis of the $\alpha$-root basis. Hence $p_S(\delta)=H^2/2+H$. Also $\rho=\alpha/2$, hence $\rho(\diag(t,t^{-1}))=t$. So $d\rho(H)=1$, $\delta_S(H)=H-1$. Therefore $$\gamma(\Delta)=\delta_S(H^2/2+H)=(H-1)^2/2+(H-1)=(H^2-1)/2\in U(\mathfrak{t})=\mathbb{C}[H].$$ The Weyl group $W\cong S_2$ and acts by $w(\diag\{t,t^{-1}\})=\diag\{t^{-1},t\}$ and thus $w(H)=-H$. Hence $\mathbb{C}[H]^W=\mathbb{C}[H^2]$. By the theorem, we know that $Z(\mathfrak{g})=\mathbb{C}[\gamma(\Delta)]=C[H^2]$.
Definition 72 Now we come back to the situation that $G$ is a reductive group over $\mathbb{R}$. Let $V$ be an irreducible admissible $(\mathfrak{g},K)$-module. A version of Schur's lemma says that every element of $Z(\mathfrak{g})$ acts as a scalar on $V$. Therefore it defines an algebra homomorphism $\chi_V: Z(\mathfrak{g})\rightarrow \mathbb{C}$, the infinitesimal character of $V$. This does not determine the representation $V$ uniquely but does tell a lot of information: e.g., the infinitesimal character of an automorphic representation knows about the Hodge-Tate weights of the associated Galois representation.
Definition 73 For $\lambda\in \mathfrak{t}^*$, we define $\chi_\lambda: Z(\mathfrak{g})\rightarrow \mathbb{C}$ by composing $\gamma: Z(\mathfrak{g})\rightarrow U(\mathfrak{t})^W$ and $\lambda: U(\mathfrak{t})\rightarrow \mathbb{C}$. It is a fact that every homomorphism $\chi: Z(\mathfrak{g})\rightarrow \mathbb{C}$ arises in this manner and $\chi_\lambda=\chi_{\lambda'}$ if and only if $\lambda,\lambda'$ are in the same $W$-orbit.
Remark 63 Suppose $G(\mathbb{R})$ is compact. We have seen a bijection between irreducible representations of $G(\mathbb{R})$ and the set of dominant weights. A nice computation shows that if $\pi$ has highest weight $\lambda$, then the infinitesimal character is $\chi_{\lambda+\rho}$. Notice it is not true for general non-compact groups.

TopSquare-integrable representations of real reductive groups

Assume $G$ is semisimple group over $\mathbb{R}$ for simplicity in this section (everything said below are true for any reductive group after appropriate modification).

Theorem 25 (Harish-Chandra) The following statements are equivalent.
  1. $G$ has a maximal tours $T$ such that $T(\mathbb{R})$ is compact.
  2. $G$ has an inner form $G'$ such that $G'(\mathbb{R})$ is compact.
  3. $G(\mathbb{R})$ has square-integrable representations.
Example 32 $SL_2$ has a compact inner form $SU(2)$, therefore $SL_2(\mathbb{R})$ has discrete series. On the other hand, for $n\ge3$, $SL_n(\mathbb{R})$ has no compact inner form (there are only two division algebra over $\mathbb{R}$!), thus has no discrete series.
Definition 74 We write $\tilde G$, the unitary dual, for the set of unitary equivalence classes of unitary irreducible admissible representations of $G(\mathbb{R})$. We write $\tilde G_d\subseteq \tilde G$ for the set of discrete series representations.

Assume $\tilde G_d$ is non-empty. Then by the previous theorem, there exists a maximal torus $T\subseteq G $ and a maximal compact $K\subseteq G(\mathbb{R})$ such that $T(\mathbb{R})\subseteq K\subseteq G(\mathbb{R})$ and $T(\mathbb{R})$ is compact.

Definition 75 The real Weyl group is defined to be $W_\mathbb{R}=N_{G(\mathbb{R})}T(\mathbb{R})/T(\mathbb{R})$. $W_\mathbb{R}$ naturally embeds into the complex Weyl group $W=W(G_\mathbb{C}, T_\mathbb{C})$.
Definition 76 We say $\mu\in X^\cdot(T_\mathbb{C})$ is regular if $\langle\mu,\alpha^\vee\rangle\ne0$ for any $\alpha\in \Phi(G_\mathbb{C}, T_\mathbb{C})$. Notice the regular condition is invariant under the action of $W$.

We have the following nice parametrization of square-integrable representations.

Theorem 26 (Harish-Chandra)
  1. If $\pi$ is a unitary square-integrable representation of $G(\mathbb{R})$, then $\pi$ has infinitesimal character $\chi_\mu$ for some $\mu\in X^\cdot(T_\mathbb{C})$ regular.
  2. There exists a bijection between the square-integrable representations of $G(\mathbb{R})$ with infinitesimal character $\chi_\mu$ and the set of $W_\mathbb{R} $-orbits inside $W\cdot \mu$. In particular there are $\#W_\mathbb{R}\backslash W$ such representations.
Example 33 If $G(\mathbb{R})$ is compact, then any representation of $G(\mathbb{R})$ is square-integrable. The theorem implies that the irreducible representations of $G(\mathbb{R})$ are in bijection with the $W$-orbits of regular elements in $X^\cdot(T_\mathbb{C})$. On the other hand, it is also in bijection with the set of dominant weights $\lambda\in X^\cdot(T_\mathbb{C})$. What is happening here?

Suppose $\pi$ is an irreducible representation of $G(\mathbb{R})$ with highest weight $\lambda$, then $\pi$ has infinitesimal character $\chi_\mu$, where $\mu=\lambda+\rho$. The $\rho$-shift $\lambda\mapsto\lambda+\rho$ indeed gives a bijection between dominant weights (dependent on a choice of the root basis) and the set of $W$-orbits of regular elements of $X^\cdot(T_\mathbb{C})$ (independent on a choice of the root basis).

Example 34 Let $U(p,q)$ be the unitary group of the Hermitian form $\langle z,w\rangle=\sum_{i=1}^p z_i\overline{w_i}-\sum_{i=p+1}^n z_i\overline{w_i}$, where $n=p+q$. The subgroup $SU(p,q)$ of determinant 1 is a semisimple group over $\mathbb{R}$, which is a form of $SL_n$ and an inner form of the compact group $SU(n)$. A compact maximal torus in $SU(p,q)$ is the diagonal torus. A maximal compact subgroup is $S(U(p)\times U(q))(\mathbb{R})$. We have $W=S_n$ and $W_\mathbb{R}=S_p\times S_q$. The packet of square-integrable representations with same infinitesimal character then has $n!/p!q!={n \choose p}$ elements.

TopRepresentations of $SL_2(\mathbb{R})$

Let $G=SL_2$, $K=SO(2)(\mathbb{R})\subseteq G(\mathbb{R})$. We have a maximal torus $T \subseteq G$ such that $T(\mathbb{R})=K$. We have $W=S_2$ and $W_\mathbb{R}=1$. Therefore the discrete series representations of $SL_2(\mathbb{R})$ fall into packets of 2 elements, parametrized by the $W$-orbits on the regular elements $\mathbb{Z}-\{0\}\subseteq X^\cdot(T_\mathbb{C})$.

Let $S\subseteq G $ be the diagonal split maximal torus, $M=\{\pm1\}\subseteq G(\mathbb{R})$, $A=\{\diag\{t,t^{-1}\}: t\in \mathbb{R}_{>0}\}$ (the connected center of $S$), $B\subseteq G$ the standard Borel and $N\subseteq B$ the unipotent radical. Then we have $S(\mathbb{R})=M\times A$ and the Langlands decomposition $B(\mathbb{R})=S(\mathbb{R})\times N(\mathbb{R})=M\times A\times N(\mathbb{R})$. The characters of $S(\mathbb{R})$ are then indexed by $(p,s)\in \{\pm1\}\times \mathbb{C}\cong \mathbb{Z}/2\times\mathbb{C}$, i.e. $\chi_{p,s}(\diag\{t,t^{-1}\})=\sgn(t)^p|t|^{s}$. The parabolic induction in this case gives a representation $\mathcal{P}^{\pm,s}$ of $G(\mathbb{R})$.

Definition 77 Consider the space $P^{\pm,s}$ of smooth functions $f: G(\mathbb{R})\rightarrow \mathbb{C}$ such that $f(bg)=\chi_{\pm,s+1}(t)f(g)$, for $b=tn\in B(\mathbb{R})$, $g\in G(\mathbb{R})$. Recall the Iwasawa decomposition $S(\mathbb{R})\times N(\mathbb{R})\times K= G(\mathbb{R})$. Therefore $f$ is uniquely determined by its restriction on $K$. We define an inner product on $P^{\pm,s}$ by $$\langle f,g\rangle=\int_{K}f(k)\overline{g(k)}dk.$$ We define $\mathcal{P}^{\pm,s}$ to be the Hilbert space completion of $P^{\pm,s}$.
Remark 64 One can show the action of $G(\mathbb{R})$ extends to $\mathcal{P}^{\pm,s}$. Moreover, if $\chi_{\pm,s}$ is unitary, then $\mathcal{P}^{\pm,s}$ is also unitary (for this we need the shift $s+1$ in the previous definition).
Remark 65 Every irreducible admissible representation of $G(\mathbb{R})$ is infinitesimally equivalent to a subquotient of $\mathcal{P}^{\pm,s}$. The Casmir element $\Delta$ acts on $\mathcal{P}^{\pm,s}$ by the scalar $(s^2-1)/2$.
Theorem 27 The irreducible admissible representations of $SL_2(\mathbb{R})$ up to infinitesimal equivalence are classified into the following types.
  1. For any $n\ge1$, there is a pair $D_n^{\pm}$ (discrete series representation). These subrepresentations of $\mathcal{P}^{(-1)^{n+1},n}$
  2. A pair $D_0^{\pm}$ (limit of discrete series representation). In fact $P^{-,0}=D_0^+\oplus D_0^-$.
  3. $P^{+,it}$ ($t\in \mathbb{R}_{\ge0}$) and $P^{-,it}$ ($t\in \mathbb{R}_{>0}$).
  4. Finite dimensional representation of dimension $n $. These are quotients of $\mathcal{P}^{(-1)^{n+1},n}$.
  5. $\Re s>0$ and either $s\not\in \mathbb{Z}$, or $s\in \mathbb{Z}$ and $p\ne(-1)^{s+1}$.

Among these the unitary ones are: a)-c), the trivial representation in d), $s\in (0,1)$ in e) (complementary series representations); the tempered ones are: a)-c); the square-integrable ones are: a).


TopThe trace formula for compact quotients

Announcement: Dick Gross will tell us about $L $-groups next Thursday (Oct 31). There will be no class on Nov 7.

The trace formula is a good tool for constructing interesting automorphic representations. We will talk about the trace formula for the compact quotients today and the simple trace formula for non-compact quotients next time.

Definition 78 Let $H $ be a (separable) Hilbert space. We say a linear operator $T: H\rightarrow H$ is Hilbert-Schmidt if for some (equivalently, any) orthonormal basis $\{e_i\}_{i=1}^\infty$ the sum $\sum_{i=1}^\infty ||T e_i||^2$ is convergent. We say $T$ is of trace class if there exists Hilbert-Schmidt operators $A $, $B $ such that $T=A^*B$.
Remark 66 If $T$ is of trace class, then the sum $\Tr T=\sum_{i=1}^\infty \langle T e_i,e_i\rangle=\sum_{i=1}^\infty\langle B e_i,Ae_i\rangle$ is absolutely convergent (by Cauchy-Schwarz) and is independent of the choice of the orthonormal basis.
Example 35 Let $(X,\mu)$ be a Hausdorff compact measure space and $K:X\times X\rightarrow \mathbb{C}$ is a continuous function, then we define $T: L^2(X)\rightarrow L^2(X)$ by the formula $$T(f)(x)=\int_{X}K(x,y)f(y)d\mu,$$ by integrating the kernel function $K$, then one can easily check that $T$ is Hilbert-Schmidt (by writing it down using an orthonormal basis).

Now let $G$ be a unimodular locally compact Hausdorff topological group. Let $\Gamma\subseteq G$ be a discrete cocompact subgroup.

Example 36 $G=\mathbb{R}$ and $\Gamma=\mathbb{Z}$.
Example 37 $G=SL_2(\mathbb{R})$ and $\Gamma$ a discrete cocompact subgroups, e.g., from a quaternion algebra over $\mathbb{Q}$ or from a compact Riemann surface uniformized by the upper half plane.

Let $X=\Gamma\backslash G$ and $H=L^2(X)$. Then $H $ is a unitary representation of $G$ under the right translation.

Definition 79 Choose $\phi\in C_c(G)$ a continuous function with compact support. We can turn this function into an operator on $H $ by defining $T_\phi:H\rightarrow H$ by $$T_\phi(f)(x)=\int_G\phi(g)f(xg)dg.$$ Then $T_\phi$ is given by integrating over $X$ the kernel function $K$ on $X\times X$: $$K_\phi(x,y)=\sum_{\gamma\in \Gamma}\phi(x^{-1}\gamma y).$$
Remark 67 If $x,y\in G$ and $U_x,U_y\subseteq G$ are compact neighborhoods, then the sum defining $K_\phi(x',y')$ can be taken, for $x'\in U_x$, $y'\in U_y$, over $\gamma\in\Gamma\cap (U_x^{-1}\cap \mathrm{supp}(\phi)\cdot U_y)$, which is simply a finite set!.
Lemma 8
  1. For any $\phi\in C_c(G)$, $T_\phi$ is Hilbert-Schmidt.
  2. Define $\phi^*(g)=\overline{\phi(g^{-1})}$. Then $T_\phi^*=T_{\phi^*}$.
  3. Define the convolution on $C_c(G)$ by $$\phi*\psi(g)=\int_G\phi(x)\psi(x^{-1}g)dx.$$ Then $T_\phi\circ T_\psi=T_{\phi*\psi}$.
Theorem 28 We can decompose $L^2(\Gamma\backslash G)\cong\hat\bigoplus_{\pi \in \hat G} m(\pi)\pi$, where each $m(\pi)$ is finite and $\hat G$ is the set of unitary equivalence classes of irreducible unitary representations of $G$.
Proof (Sketch) We choose $\phi\in C_c(G)$ such that $\phi^*=\phi$. Thus $T_\phi$ is self-adjoint. Since $T_\phi$ is also Hilbert-Schmidt, it is compact. Now applying the spectral theorem for compact operators on Hilbert spaces implies that $$H=H_{\phi,0}\oplus \hat \bigoplus H_{\phi, \lambda},$$ where $H_{\phi,\lambda}$ is the $\lambda$-eigenspace for $T_\phi$ and each of these is finite dimensional.

If $V\subseteq H$ is the closed linear span of all $H_{\phi,\lambda}$ as $\phi=\phi^*$ varies and $\lambda\ne0$ varies,then $V=H$. Indeed, if $f\in V^\perp$, then for all $\phi=\phi^*$, $T_\phi(f)=0$. But one can approximate the identity operator by $T_\phi$, hence $f=0$.

Though $L^2(\Gamma\backslash G)$ is not admissible as a $G$-representation in general, we can still run a similar argument as Lemma 3 using the fact that $H_{\phi,\lambda}$ is finite dimensional. We claim that $H $ has a closed irreducible $G$-invariant subspace. Choose $H_{\phi,\lambda}\ne0$. Choose $W'\subseteq H$ a closed $G$-invariant subspace such that $W'\cap H_{\phi,\lambda}$ is non-zero and is minimal with that property. Let $W$ be the intersection of all closed $G$-invariant subspace $W''$ of $H $ such that $W''\cap H_{\phi,\lambda}=W'\cap H_{\phi,\lambda}$. Then $W$ is irreducible. Indeed if $W=W_1 \oplus W_2$, then since $T_\phi$ is self-adjoint and any $G$-invariant subspace is also $T_\phi$-invariant, we know that $(W_1 \oplus W_2)\cap H_{\phi,\lambda}=(W_1 \cap H_{\phi,\lambda}) \oplus (W_2 \cap H_{\phi,\lambda})$. The construction of $W$ tells us that either $W_1=0$ or $W_2=0$.

Now the decomposition follows by applying the same argument to $W^\perp$. The finiteness of $m(\pi)$ follows also from the fact that $H_{\phi,\lambda}$ is finite dimensional.

Remark 68 If $\phi \in C_c(G)$ is a convolution (or a sum of convolutions), then $T_\phi$ is clearly of trace class.
Lemma 9 If $T_\phi$ is of trace class, then $$\tr T_\phi=\int_{\Gamma\backslash G}K(x,x)dx.$$
Proof Assume that $\phi=\phi^*$ (in general, we can write $\phi=(\phi+\phi^*)/2+i\cdot (\phi-\phi^*)/2i$, a linear combination of two self-adjoint functions). We can choose orthonormal eigenvectors $e_i$ of $T_\phi$ forming an orthonormal basis of the subspace $\hat\bigoplus_{\lambda\ne0} H_{\phi,\lambda}\subseteq H$. These functions $e_i\in L^2(\Gamma\backslash G)$ are continuous as they are the images of $T_\phi$. We can write $$K(x,y)=\sum_{n\ge1}\lambda_n e_n(x)\overline{e_n(y)}$$ for $\lambda_n\in \mathbb{C}$, where the sum converges in $L^2(\Gamma\backslash G\times \Gamma\backslash G)$. Write $K_N$ be the truncated sum of $K$ up to $N $. Since $T_\phi$ is of trace class, we know that $||K_N-K||_{L^1(\Gamma\backslash G)}\le\sum_{n> N}|\lambda_n|\rightarrow0$. We have $\int_{\Gamma\backslash G}K_N(x,x)dx=\sum_{n\le N}\lambda_n$ and $$\tr T_\phi=\lim_{N\rightarrow \infty}\int_{\Gamma\backslash X}K_N(x,x)dx=\int_{\Gamma\backslash G}K(x,x)dx$$ since $K_N\rightarrow K$ in $L^1(\Gamma\backslash G)$.

The trace formula relates two different expressions (the geometric side and the spectral side) for $\tr T_\phi$ whenever this makes sense.

The geometric side is given by (using the previous lemma) 
  \tr T_\phi&=\int_{\Gamma\backslash G}\sum_{\gamma\in\Gamma}\phi(x^{-1}\gamma x)dx\\
  &  =\int_{\Gamma\backslash}\sum_{\gamma\in\{\Gamma\}}\sum_{\delta\in \Gamma_\gamma\backslash \Gamma}\phi(x^{-1}\delta^{-1}\gamma\delta x)dx\\
  & =\sum_{\gamma\in\{\Gamma\}}\int_{\Gamma_\gamma\backslash G}\phi(x^{-1}\gamma x)dx\\
  &=\sum_{\gamma\in\{\Gamma\}}\Vol(\Gamma_\gamma\backslash G_\gamma)\int_{G_\gamma\backslash G}\phi(x^{-1}\gamma x).
Here $\{\Gamma\}$ means the conjugacy classes in $\Gamma$. $\Gamma_\delta$ is the centralizer of $\gamma$ in $\Gamma$, $G_\gamma$ is the centralizer of $\gamma$ in $G$. The quotient measure makes sense since $G_\gamma$ is unimodular.

The spectral side is much simpler at this stage: $$\Tr T_\phi=\sum_{\pi\in \hat G} m(\pi)\tr\pi(\phi).$$

Theorem 29 If $T_\phi$ is of trace class, then $$\sum_{\pi} m(\pi)\tr \pi(\phi)=\sum_{\gamma\in\{\Gamma\}}a(\gamma)\mathcal{O}_\gamma(\phi),$$ where $a(\gamma)=\Vol(\Gamma_\gamma\backslash G_\gamma)$ and $\mathcal{O}_\gamma(\phi)=\int_{G_\gamma\backslash G}\phi(x^{-1}\gamma x)dx$.
Remark 69 The quantities $a_\gamma$ and $\mathcal{O}_\gamma(\phi)$ depend on the choices of measures on $G$, $\Gamma_\gamma\backslash G_\gamma$ and $G_\gamma\backslash G$. We assume they are chosen compatibly. The product $a(\gamma)\mathcal{O}_\gamma(\phi)$ is independent of the choices.

Now assume $F$ is a number field and $G/F$ is a semisimple anisotropic (= contains no nontrivial split torus) group. In this case, the quotient $G(F)\backslash G(\mathbb{A}_F)$ is compact.

Definition 80 Let $C_c^\infty(G(\mathbb{A}_F))=C_c^\infty(G(\mathbb{A}_F^\infty)) \otimes_\mathbb{C} C_c^\infty(G(F \otimes_\mathbb{Q}\mathbb{R}))$ be the space of locally constant/smooth and compactly supported functions. By a nice theorem of Dixmier-Malliavin, every function $\phi\in C_c^\infty(\mathbb{A}_F)$ is a (finite sum of) convolutions (this is trivial on $G(\mathbb{A}_F^\infty)$ by convolving an indicate function on the support, but is far from trivial on $G(F \otimes_\mathbb{Q}\mathbb{R})$). In particular $T_\phi$ is of trace class for such a function $\phi$.
Proposition 21
  1. Let $K_\infty\subseteq G(F \otimes_\mathbb{Q} \mathbb{R})$ be a maximal compact subgroup. If $(\pi,H)$ is an irreducible unitary representations of $G(\mathbb{A}_F)$. Them the submodule $V\subseteq H$ of $K$-finite vectors for $K=K^\infty K_\infty$ (any $K^\infty\subseteq G(\mathbb{A}_F^\infty)$ compact open subgroup) is an algebraically irreducible admissible $(\mathfrak{g},K)\times G(\mathbb{A}_F^\infty)$-module.
  2. The above assignment $H\mapsto V$ defines a bijection between the unitary equivalence classes of irreducible unitary representations of $G(\mathbb{A}_F)$ and the isomorphism classes of unitary irreducible admissible $(\mathfrak{g},K)\times G(\mathbb{A}_F^\infty)$-modules.
  3. The above assignment $H\mapsto V$ restricts to a bijection between the unitary equivalence classes of irreducible representations $\pi$ which appear in $L^2(G(F)\backslash G(\mathbb{A}_F))$ and the isomorphism classes of automorphic representations of $G(\mathbb{A}_F)$.
Proof See Flath in Corvallis I.
Remark 70 Part c) really depends on the fact that $G$ is anisotropic: integrating over the compact quotient immediately gives the unitary structure. In general, there is an analogue of part c) for cuspidal automorphic representations, but not for all automorphic representations (Theorem 30).


TopThe simple trace formula

References for this section:

  • Henniart, Mem. Soc. Math. France 1984;
  • Arthur, CJM 1986;
  • Arthur, JAMS 1988.

In this section we let $F$ be a number field and $G/F$ be a semisimple group (it is easy to extend the results below to the case where $G$ is reductive).

Recall form last time that if $\phi\in C_c^\infty(G(\mathbb{A}_F))$ and $G$ is anisotropic, then we gave a formula expressing $\tr T_\phi$ in two ways $$\sum_\pi m(\pi)\tr\pi(\phi)=\sum_{\gamma\in\{G(F)\}}a(\gamma)\mathcal{O}_\gamma(\phi).$$ The spectral side sums over (countably many) automorphic representations of $G(\mathbb{A}_F)$. The geometric side sums over the conjugacy classes of $G(F)$.

Such a formula does not exist if $G$ is not anisotropic (e.g., a group as simple as $PGL_2$). The two main problems are

  1. $L^2(G(F)\backslash G(\mathbb{A}_F))$ does not decompose discretely as the Hilbert direct sum $\hat\bigoplus_{i\in I} \pi_i$. There exists a certain continuous spectrum.
  2. The kernel function $K(x,x)=\sum_{\gamma\in G(F)}\phi(x^{-1}\gamma x)$ on $G(F)\backslash G(\mathbb{A}_F)$ is no longer integrable. The operator $T_\phi$ is no longer of trace class.

Here are two possible ways out:

  1. Describe the continuous spectrum and truncate $K$ to get a well-defined expression. This was done by Arthur after incredible amount of work and the formula thus obtained is called the Arthur trace formula.
  2. Adopt some simplifying assumptions on the allowable test functions $\phi$. We shall consider this direction in the sequel.
Definition 81 We define the cuspidal subspace $L_0^2(G(F)\backslash G(\mathbb{A}_F))\subseteq L^2(G(F)\backslash G(\mathbb{A}_F))$ as the subspace of the functions $f$ such that for all parabolics $P=MN\subseteq G$ defined over $F$, the integral $$\int_{N(F)\backslash N(\mathbb{A}_F)}f(gn)dn=0$$ for almost every $g\in G(\mathbb{A}_F)$. We only require "almost every" since there is no reason the integral always makes sense.
Theorem 30
  1. The subspace $L_0^2(G(F)\backslash G(\mathbb{A}_F))\subseteq L^2(G(F)\backslash G(\mathbb{A}_F))$ is closed, $G(\mathbb{A}_F)$-invariant and decomposes discreetly as a countable Hilbert direct sum of subrepresentations, each appearing with finite multiplicity. For $\phi\in C_c^\infty(G(\mathbb{A}_F))$, $T_{0,\phi}$, the restriction of $T_\phi$ on $L_0^2$, is of trace class.
  2. The assignment $H\mapsto V$, where $V$ is the $K$-finite vectors in $H $, gives a bijection between the set of unitary equivalence classes of irreducible subrepresentations of $L_0^2$ and the set of cuspidal automorphic representations of $G(\mathbb{A}_F)$.
Definition 82 Suppose $k$ is any field of characteristic 0, $H $ is a reductive group over $k$. Let $\gamma\in H(k)$.
  1. We say $\gamma$ is elliptic if $\gamma$ is contained in an elliptic maximal torus of $H $ (so $\gamma$ is semisimple). A torus $T$ of $H $ is called elliptic (= anisotropic) if $X^\cdot(T_{\bar k})^{\Gamma_k}=0$. When $k=\mathbb{R} $, a torus is elliptic if and only if its real points is compact. So the identity $e\in G(\mathbb{R})$ is elliptic if and only if $G_\mathbb{R}$ contains a compact torus, if and only if $G(\mathbb{R})$ has discrete series representations.
  2. We say $\gamma$ is regular semisimple, if $Z_H(\gamma)^0$ is a maximal torus. For $G=SL_n$, $\gamma\in H(k)$ is regular semisimple if and only if $\gamma$ is semisimple and its characteristic polynomial has distinct roots; $\gamma$ is elliptic regular semisimple if and only if $\gamma$ is semisimple and its characteristic polynomial has distinct roots and is irreducible over $k$.
Theorem 31 Let $\phi=\prod_v\phi_v\in C_c^\infty( G(\mathbb{A}_F))$. Suppose $\phi$ satisfies that there exists places $v_1\ne v_2$ of $F$ such that
  1. $\phi_{v_1}$ is a matrix coefficient of a supercuspidal representation of $G(F_{v_1})$;
  2. For any $\gamma\in G(F_{v_2})$, $\mathcal{O}_\gamma(\phi_{v_2})=0$ unless $\gamma$ is $F_{v_2}$-elliptic.

Then $$\sum_\pi m_0(\pi)\tr\pi(\phi)=\sum_{\gamma\in \{G(F)\},\gamma\text{ elliptic}}a(\gamma)\mathcal{O}_\gamma(\phi),$$ where the sum on the spectral side is over the (countably many) cuspidal automorphic representations of $G(\mathbb{A}_F)$ and $m_0(\pi)$ is the multiplicity of $\pi$ in $L_0^2(G(F)\backslash G(\mathbb{A}_F))$.

Theorem 32 Let $\phi=\prod_v\phi_v\in C_c^\infty( G(\mathbb{A}_F))$. Suppose $\phi$ satisfies that there exists places $v_1\ne v_2$ of $F$ such that
  1. $\phi_{v_1}$ is a matrix coefficient of a supercuspidal representation of $G(F_{v_1})$;
  2. For any $\gamma\in G(F_{v_2})$, $\phi(\gamma)=0$ unless $\gamma$ is regular $F_{v_2}$-elliptic (this condition is strictly stronger than part b of Theorem 31).

Then $$\sum_\pi m_0(\pi)\tr\pi(\gamma)=\sum_{\gamma\in \{G(F)\}}a(\gamma)\mathcal{O}_\gamma(\phi).$$

Remark 71 Theorem 31 is due to Clozel, Kottwitz and Arthur. See Arthur JAMS, 88, Section 7. Theorem 32 is proved earlier by Deligne-Kazhdan. See Henniart MSMF, 84 for a proof. Any formula of this type (by imposing certain conditions on the the test functions) is called a simple trace formula.
Remark 72
  1. Theorem 31 is much stronger than Theorem 32. It requires the knowledge only about the orbital integrals $\mathcal{O}_\gamma(\phi)$, not about the function $\phi$ itself. Indeed the space of orbital integrals is dense in the space of all invariant distributions and the conditions $\mathcal{O}_\gamma(\phi)=0$ says nothing about the support of $\phi$. In applications, we know more about the orbital integrals (e.g., matching from transfers) than the function $\phi$ itself, so Theorem 31 is much more powerful than Theorem 32. The proof of Thereom 31 needs the full power of Arthur's invariant trace formula.
  2. Theorem 31 allows the term form singular elliptic element of $G(F)$. For example, the identity class $1\in G(F)$ is elliptic but singular. We have $a(1)\mathcal{O}_1(\phi)=\Vol(G(F)\backslash G(\mathbb{A}_F))\phi(1)$, which is closely related to Weil's conjecture on Tamagawa numbers. For any $G$ semisimple, $G(F)\backslash G(\mathbb{A}_F)$ has finite volume. There is a canonical measure, the Tamagawa measure $\tau$, induced by an algebraic volume form on $G$. Weil conjectured that the Tamagawa number $\tau(G(F)\backslash G(\mathbb{A}_F))$ is equal to one for $G$ simply-connected. Langlands proved this conjecture for split $G$ and suggested a proof by comparison of the trace formula for general $G$. Kottwitz proved the conjecture by exactly using Theorem 31, in Annals 1988.

The proof Theorem 32 is much easier than Theorem 31. We shall sketch a proof of Theorem 32.

Proof (Sketch)
Lemma 10 The image of $T_\phi\subseteq L_0^2(G(F)\backslash G(\mathbb{A}_F))$.
Proof This only uses the assumptions a) on $v_1$: $\phi_{v_1}$ is a matrix coefficient of a supercuspidal representation. We need to use: if $f\in C_c^\infty(G(\mathbb{A}_F))$ is a supercuspidal matrix coefficient, then for any parabolic $P=MN\subseteq G_{F_{v_1}}$ defined over $F_{v_1}$, we have $$\int_{N(F_{v_1})}f(xny)dn=0$$ for any $x,y\in G(F_{v_1})$. We take $f\in L^2(G(F)\backslash G(\mathbb{A}_F))$, then $$T_\phi(f)(g)=\int_{G(\mathbb{A}_F)}\phi(h)f(gh)dh.$$ Choose a parabolic $P=MN\subseteq G$ defined over $F$. To check the cuspidality, we need to compute the integral 
\int_{N(F)\backslash N(\mathbb{A}_F)} T_\phi f(ng)dn&=\int_{N(F)\backslash N(\mathbb{A}_F)}\int_{G(\mathbb{A}_F}\phi(h)f(ngh)dhdn\\&=\int_{N(F)\backslash N(\mathbb{A}_F)}\int_{N(F)\backslash G(\mathbb{A}_F)}\sum_{\gamma\in N(F)}\phi(g^{-1}n^{-1}\gamma h)f(h)dhdn.
Now $N(F)\backslash N(\mathbb{A}_F)$ is compact and $\phi$ is compactly supported, so we can reverse the order of integration to obtain $$\int_{N(F)\backslash N(\mathbb{A}_F)}\int_{N(\mathbb{A}_F)}\phi(g^{-1}g^{-1}h)dnf(h)dh.$$ One can choose the measure on $N(\mathbb{A}_F)$ to be a product measure $dn=\prod dn_v$, the assumption on $\phi_{v_1}$ shows that the contribution from the $v_1$-factor is zero. Hence the above integral itself is zero.

This lemma implies that $T_\phi$ is of trace class and $\tr T_\phi=\tr T_{0,\phi}$. One can then show that the kernel function $K_\phi(x,x)=\sum_{\gamma\in G(F)}\phi(x^{-1}\gamma x)$ is integrable along $G(F)\backslash G(\mathbb{A}_F)$ and this integral equals to $\tr T_\phi$.

We need another lemma which is also easy to prove, though may take us too far afield.

Lemma 11 The function $x\mapsto\sum_{\gamma\in G(F)}|\phi(x^{-1}\gamma x)|$ has compact support on $G(F)\backslash G(\mathbb{A}_F)$.
Proof This lemma uses the assumption b) on $v_2$. The proof uses the reduction theory of Arthur. Morally speaking, the elliptic classes does not end up in the cusps. See Gelbart, Lectures on the Arthur-Selberg trace formula, for an example for $SL_2$.

Consequently, one has $$\int_{G(F)\backslash G(\mathbb{A}_F)}\sum_{\gamma \in G(F)}|\phi(x^{-1}\gamma x)|dx<\infty.$$ The same manipulation as in the compact quotients is now valid and lead to Theorem 32.

Remark 73 In both theorems, the sum on the geometric side has only finitely many terms (the spectral side may have infinitely many terms). In fact there are only finitely many elliptic conjugacy classes whose $G(\mathbb{A}_F)$-conjugacy classes meet the support of $\phi$. This is another application of Arthur's reduction theory. See Arthur CJM, 1986.

Here is one application of Theorem 32.

Theorem 33 Let $G$ be a semisimple group over a number field $F$. Let $v_1$ be a place of $F$, $\pi_1$ a supercuspidal representation of $G(F_{v_1})$. Then there exists a cuspidal automorphic representation $\Pi$ of $G(\mathbb{A}_F)$ such that $\Pi_{v_1}\cong \pi_1$.
Remark 74 This theorem is used in Deligne-Kazhdan-Vigneras's proof of the local Jacquet-Langlands correspondence for $GL_n$ and the multiplicative group of a division algebra $D^\times$.
Proof (Sketch) Use Theorem 32 with $\phi=\prod_v\phi_v$, where $\phi_{v_1}$ is a matrix coefficient of $\pi_{v_1}^\vee$ and other $\phi_v$'s are arbitrary. It suffices to show that $$\sum_{\Pi}m_0(\Pi)\tr\Pi(\phi)\ne0.$$ If this holds, then there exists at least one cuspidal automorphic representation $\Pi$ such that $\tr \Pi(\phi)\ne0$. But if $\sigma$ is any irreducible admissible representation of $G(F_{v_1})$ and $\tr\sigma(\phi_{v_1})\ne0$, then $\sigma\cong\pi_{v_1}$ (one can write an explicit map from $\sigma$ to $\pi_{v_1}$). To show the above sum is nonzero, we look at the the finitely many terms on the geometric side. If the measures are chosen appropriately, then the orbital integral $\mathcal{O}_\gamma(\phi)$ factors as a product of local orbital integrals $\prod_v\mathcal{O}_{\gamma}(\phi_v)$. One can manage to choose $\phi_v$ such that exactly one class $\gamma\in\{G(F)\}$ contributes to the geometric side and $\mathcal{O}_\gamma(\phi)\ne0$.


TopLanglands dual groups

This is a guest lecture by Dick Gross. Notice the notation is different sometimes.

Reference: Casselman, Survey on $L $-groups.

For each reductive group $G$ over $k$, we are going to associate a complex Lie group ${}^{L}G$. For a torus, one simply switches the role of the character group $M=X^\cdot(T)$ and the cocharacter group $M^\vee=X_\cdot(T)$.

For a general reductive group $G$, recall that one defines the roots $\Phi\subseteq M$ using a torus $T\subseteq G $ by looking at the action of $T$ on $\Lie G$; one then defines the coroots using a $\mathfrak{sl}_2$-triple associated to the roots. This gives a natural bijection $\alpha\mapsto \alpha^\vee $ between the roots and coroots and $\langle \alpha,\alpha^\vee\rangle=2$ under the natural pairing. The root datum $R=(M,\Phi,M^\vee,\Phi)$ classifies reductive group over algebraically closed field. The simple reflections $s_\alpha$ preserves $\Phi$ hugely restricts the possibility of the root datum and the classification boils to the case of rank 2 root system $A_2$, $B_2=C_2$, $D_2$ and $G_2$.

Remark 75 In Jack's notation, $R$ stands for the based root datum, which Dick calls $BR$ below.

The problem of this classification is that the natural map $\Aut(G,T)\twoheadrightarrow\Aut(R)$ is not injective. For example, the elements $N_G(T)$ acts non-trivially on $R$. Even worse, $W=N_G(T)/T\rightarrow\Aut(R)$ is not surjective:

Example 38 Consider $G=GL_n$. Then $M=\oplus_{i=1}^n \mathbb{Z}e_i$, where $e_i$ takes a diagonal element to its $i$-th diagonal entry; $M^\vee=\oplus_{i=1}^n \mathbb{Z}e_i^\vee$, taking $\lambda$ to a diagonal element $\diag\{1,1,\ldots,\lambda,1\ldots,1\}$; $\Phi=\{e_i-e_j\}$ with root basis $g_{e_i-e_j}=E_{ij}$. The coroot $(e_i-e_j)^\vee=e_i^\vee-e_j^\vee$. The sublattice $\mathbb{Z}\Phi\subseteq M$ has corank 1. The Weyl group is $S_n$ and $s_{e_i-e_j}=(ij)$. This is not all $\Aut(R)$ because $-1\in GL(M)$ is always an automorphism of $R$! It turns out to be the case $\Aut(R)=S_n\times\{\pm1\}$.
Example 39 Another extremal case is that $G=T$ is a torus of dimension $n $, then $W=1$ but $\Aut(R)=GL_n(\mathbb{Z})$!

Now choosing a Borel subgroup $B\supseteq T$ (these are permuted by $W$ simply-transitively) gives extra structure: the set of positive roots $\Phi_+$ and the root basis $\Delta\subseteq \Phi_+$ consisting of simple roots. The Weyl group no longer acts on this based root datum $BR$. The theorem is that

Theorem 34 $\Aut(R)=W\rtimes\Aut(BR)$.
Remark 76 However, this is where people often make a mistake: in the $GL_n$ case, $\Aut(BR)$ is generated by $(1n)(2 n-1)\ldots\times (-1)$ but not $-1$: one needs to preserve the set of simple roots! (c.f., Example 13).

One can not lift the action of $W$ to an action of $W$ in a natural way: these is no canonical way to split the sequence $0\rightarrow T\rightarrow N_G(T)\rightarrow W\rightarrow 0$ but one can lift elements of $\Aut(R)$! The group preserving $T$ and $B $ is $T/Z(G)$ which is smaller than $N_G(T)/T$ but still too large. The idea is to require that it also preserves a pinning $\mathcal{P}$ (if you imagine that Borels are like the wings of a butterfly, one "pins" it down): a basis $X_\alpha$ for each $\mathfrak{g}_\alpha$. Now $T$ acts transitively on the set of pinning, and the stabilizer is exactly $Z(G)$. The upshot is that there is no inner automorphism preserving the based root datum.

Theorem 35 (Chevalley) $\Aut(BR)=\Aut(G,\mathcal{P})$ and $\Aut(G)\cong G/Z(G)\rtimes \Aut(BR)=G/Z\rtimes \Aut(G,\mathcal{P})$.

One can do the same thing for groups which are quasi-split but not necessarily split. Suppose $T$ is split over $E $. The $\Gal(E/k)$ acts on $M, M^\vee$. Chevalley noticed that switching the role of $M, M^\vee$ gives you another based root datum $BR^\vee$ with the same automorphism group $\Aut(BR^\vee)$.

Example 40 For $SL_2$, the $\Delta=\{2\}\subseteq \mathbb{Z}=M$ and $\Delta^\vee=\{1\}\subseteq \mathbb{Z}=M^\vee$ (since they have to product to 2). The dual root datum gives that of $PGL_2$. The dual root datum can be even more wired in general, for example $BR(Sp_{2n})^\vee=BR(SO_{2n+1})$.

Chevalley noticed this dual operation but didn't know what to do with it. Langlands realized what to do with it: he called it the dual group $\hat G$, a connected reduction group over $\mathbb{C}$ (the choice $\mathbb{C}$ is a bit artificial) up to isomorphism with the based root datum $BR(G)^\vee$. In particular, the Galois group $\Gal(E/k)$ acts on $\hat G$.

Langlands generalized this to define a group that is not necessarily connected.

Definition 83 The Langlands dual group ${}^{L}G=\hat G\rtimes \Gal(E/k)$, where $\Gal(E/k)$ acts through pinned automorphism of $\hat G$. We will later see taking this semidriect product is a bad idea, but this is the first thing you should do.
Example 41 For $G=GL_n$, ${}^{L}G=GL_n(\mathbb{C})$.
Example 42 For $G=U_n(E/k)$, ${}^{L}G=GL_n(\mathbb{C})\rtimes \Gal(E/k)$. What is this group? It is a subgroup of $Sp_{2n}(\mathbb{C})$ (when $n $ is even) or $O_{2n}(\mathbb{C})$ (when $n $ is odd). Take the Siegel parabolic $P=GL_n(\mathbb{C})\Sym^2(\mathrm{Std})$ (when $n $ is even) or $P=GL_n(\mathbb{C})\bigwedge^2(\mathrm{Std})$ stabilizing the maximal flag of isotropic subspaces, then the normalizer of the Levi in this Siegel parabolic is exactly the ${}^{L}G$!
Definition 84 A Langlands parameter for a local field $k$ is a homomorphism $\phi: \Gal(k^s/k)\rightarrow \hat G$, up to conjugation by $\hat G$ (to be modified later: one shall use the Weil group $W_k$ instead $\Gal(k^s/k)$ so that one can send the Frobenius to any semisimple element (not necessarily of finite order); one shall need an extra $SL_2(\mathbb{C})$-factor to account for the monodromy operator which is unipotent).
Remark 77 Consider $G=PGL_2$. Almost all interesting parameters factors through $N(\hat T)\subseteq SL_2(\mathbb{C})$, but $N(\hat T)$ is not a semidirect product, hence is not the $L $-group of $T=U_1(E/K)$. So by requiring the semidirect product we lose the possibility of inductive procedure when studying the parameters.
Definition 85 Let $Z_{\hat G }(\phi)\subseteq \hat G$ be the stabilizer of $\phi$. We define $A_\phi=\pi_0(Z_{\hat G})(\phi)$. It is a finite group attached to the Langlands parameter. $A_\phi$ is trivial for $GL_n$, but can be nontrivial for other groups.

The local Langlands conjecture says

Conjecture 1 There is a bijection between the classes of $\{\phi,\rho\}$, where $\phi: W_k\rightarrow {}^{L}G$ is a Langlands parameter and $\rho$ is a representation of the finite group $A_\phi$, and the isomorphism classes of irreducible admissible representations of $G(k)$.
Example 43 Consider $G=SL_2$ over $\mathbb{Q}_p$, $p>2$. The ${}^{L}G=PGL_2(\mathbb{C})=SO_3(\mathbb{C})$. Let $\Gamma\cong \mathbb{Z}/2 \mathbb{Z}\times \mathbb{Z}/2 \mathbb{Z}\subseteq SO_3(\mathbb{R})\subseteq SO_3(\mathbb{C})$ be the reflection around two axes. Its normalizer is $S_4=\Gamma\cdot S_3$ and its centralizer is itself. By Kummer theory, there is a unique $(2,2)$-extension of $\mathbb{Q}_p$ since $\mathbb{Q}_p^\times/(\mathbb{Q}_p^\times)^2\cong \mathbb{Z}/2 \mathbb{Z}\times \mathbb{Z}/2 \mathbb{Z}$. One obtains a unique (up to the $S_3$-conjugation) Langlands parameter $\phi$ and $A_\phi=\mathbb{Z}/2 \mathbb{Z}\times \mathbb{Z}/2 \mathbb{Z}$. It should parametrize four representations of $SL_2(\mathbb{Q}_p)$. What are they? All these representations are depth zero supercuspidal constructed by Deligne and Lusztig. There are two maximal compact subgroups up to conjugacy of $SL_2(\mathbb{\mathbb{Q}}_p)$: $SL_2(\mathbb{Z}_p)$ and its conjugate by $\diag\{p,1\}$. These four representations are exactly the induced representations from the two half discrete series representations of dimension $(p-1)/2$ of $SL_2(\mathbb{F}_p)$! The case $p=3$ is already interesting: $SL_2(\mathbb{F}_3)$ is a double covering of $A_4$ of order 24, the two half discrete series representations are of dimensional one, i.e., the two cubic character of $\mathbb{Z}/3 \mathbb{Z}$.


We are back to Jack's notation.


  • Borel, Corvallis II. It includes everything (expected to be true) about $L $-groups;
  • Langlands, Problems in the theory of automorphic forms (historical document;
  • Gross-Reeder, From Laplace to Langlands.

Let $F$ be a field of characteristic 0 and $\Gamma_F=\Gal(\overline{F}/F)$. Let $G$ be a reductive group over $F$. Let $R$ be the bases root datum of $G_{\bar F}$. Then the Galois action on the root datum $R$ of $R$ induces homomorphism $\mu_{G}: \Gamma_F\rightarrow\Aut(R)$ (which only depends on the quasi-split inner form of $G$).

We observed that $R^\vee$ by swapping the role of roots and coroots is still a based root datum and $\Aut(R^\vee)\cong\Aut(R)$. We let $\hat G$ be the split reductive group over $\mathbb{C}$ (sometimes we also use $\overline{\mathbb{Q}_\ell}$ depending on the situation) with the based root datum $R^\vee$ equipped with a pinning $\mathcal{P}$ giving rise to $R^\vee$.

Definition 86 Define ${}^{L}G=\hat G(\mathbb{C})\rtimes \Gamma_F$, where $\Gamma_F$ acts via $\mu_G$ on $\Aut(\hat G,\mathcal{P})$. If $E/F$ is a Galois extension such that $\mu_G|_{\Gamma_E}$ is trivial, people also use the definition ${}^{L}G=\hat G(\mathbb{C})\times \Gal(E/F)$.
Remark 78 In geometric Langlands there is a way of constructing the dual group without using the root datum, but not in this generality.
Remark 79 Notice the whole construction only depends the quasi-split inner form of $G$.

TopLanglands parameters

TopLocal Langlands conjecture

The introduction of the $L $-group allows us to state the Langlands conjecture. Assume now $F$ is a $p$-adic field, $\mathbb{R}$ or $\mathbb{C}$.

Definition 87 If $F$ is $p$-adic, we define a Langlands parameter to be continuous homomorphism $W_F\times SL_2(\mathbb{C})\rightarrow {}^{L}G$, where ${}^{L}G$ is endowed with usual analytic topology, satisfying
  1. The composite homomorphism $W_F\rightarrow {}^{L}G\rightarrow\Gamma_F$. is the usual inclusion.
  2. For any $g\in W_F$, $\phi(g)$ is semisimple (i.e., for every representation of ${}^{L}G$ factoring through a finite quotient of $\Gamma_F$, $g$ acts semisimply). Notice it is not equivalent to saying that the projection onto the two factors are semisimple (product of two non-commuting semisimple elements may not be semisimple).
  3. The restriction $\phi|_{SL_2(\mathbb{C})}: SL_2(\mathbb{C})\rightarrow \hat G(\mathbb{C})$ is induced by an algebraic homomorphism $SL_2\rightarrow\hat G$.

We say two Langlands parameters $\phi$, $\phi'$ are equivalent if they are $\hat G(\mathbb{C})$-conjugate. We write $\Phi(G)$ for the set of classes of Langlands parameters of $G$. Let $\Irr_G$ be the set of isomorphism classes of irreducible admissible representations of $G(F)$.

Definition 88 If $F=\mathbb{R}$ or $\mathbb{C}$, a Langlands parameter is a semisimple continuous homomorphism (i.e., a direct sum of irreducible ones, for any representation of ${}^{L}G$) $\phi: W_F\rightarrow {}^{L}G$ such that $W_F\rightarrow {}^{L}G \rightarrow\Gamma_F$ is the usual map. As the $p$-adic case, we define similarly the notion of equivalence.

Let $\Phi(G)$ be the classes of Langlands parameters of $G$ and $\Irr_G$ be the set of isomorphism classes of irreducible admissible $(\mathfrak{g},K)$-module, where $\mathfrak{g}=\Lie(G(F)) \otimes_\mathbb{R}\mathbb{C}$ and $K\subseteq G(F)$ is a maximal compact subgroup.

Conjecture 2 (Local Langlands conjecture) There is a natural partition $\Irr_G=\coprod_{\phi\in \Phi(F)}\Pi_\phi$ into finite disjoint sets $\Pi_\phi$. If $G$ is quasi-split, every $\Pi_\Phi$ is non-empty. The set $\Pi_\phi$ is called a $L $-packet.

Of course this conjecture has no content without assuming extra condition characterizing the $L $-packets $\Pi_\phi$. The following cases are known:

  1. When $F=\mathbb{R},\mathbb{C}$, the conjecture is known and the set $\Pi_\phi$ can be written down explicitly using the parametrization of $\hat G_d$ as a starting point. This is due to Langlands.
  2. If $F$ is $p$-adic and $G$ is a torus, then the correspondence exists and can be essentially constructed from local class field theory. If $G=GL_1$, this is nothing but local class field theory.
  3. When $F$ is $p$-adic and $G=GL_n$. In this case, all the sets $\Pi_F$ has size one and gives a bijection between $\Irr_G$ and $\Phi(G)$. This is due to Harris-Taylor, Henniart. Scholze recently gave a new proof. The correspondence is characterized by some compatibility condition arising from the theory of $L $-functions due to Henniart. Henniart showed there is at most one of such bijection. Harris-Taylor proved the existence of such a bijection satisfying this characterization.
  4. When $F$ is $p$-adic and $G$ is a quasi-split classical group. The correspondence exists and is characterized by comparison with $GL_n$ due to the recent work of Arthur and Mok. This is established using the fact that these groups are twisted endoscopy groups of $GL_n$ and the method of twisted trace formula (including the fundamental lemma).
  5. If $F$ is $p$-adic and $G$ is unramified, then there is a natural correspondence between the unramified elements in $\Irr_G$ and the unramified $L $-packets (namely these factor through $W_F\rightarrow \mathbb{Z}\rightarrow {}^{L}G$).

TopUnramified local Langlands correspondence

Let $F$ be a $p$-adic field and $G/F$ be an unramified reductive group. Fix $K\subseteq G(F)$ a hyperspeicial maximal compact subgroups.

Theorem 36 There is a canonical bijection between $\pi\in\Irr_G$ such that $\pi^K\ne0$ and unramified parameters $\phi\in \Phi(G)$.
Proof Let $S\subseteq G $ be a maximal split torus. The centralizer $T=Z_G(S)$ is a maximal torus of $G$ (Definition 52). The group $W_d=N_G(S)/T$ is a constant group scheme over $F$ and acts faithfully on $S$. Let $E/F$ be the minimal extension splitting $G$. Then $E/F$ is unramified. Let $\Frob$ be the arithmetic Frobenius and $\mathrm{Fr}\in \Gal(E/F)$ be its image. The ${}^{L}G=\hat G(\mathbb{C})\times \Gal(E/F)$ contains the subset $\hat G(\mathbb{C})\times\{\mathrm{Fr}\}$, which is normalized by $\hat G(\mathbb{C})$. If $\phi$ is an unramified Langlands parameter. Then it is determined by $\phi(\Frob)\in \hat G(\mathbb{C})\times\{\mathrm{Fr}\}$, i.e., semisimple $\hat G(\mathbb{C})$-conjugacy classes in $\hat G(\mathbb{C})\times\{\mathrm{Fr}\}\subseteq {}^{L}G$.

On the other hand, the unramified represetnations are parametrized by $\hat S(\mathbb{C})/W_d$ by the Satake isomorphism (Corollary 6).

Since $W_d$ acts on $S$, $T$ hence on $\hat T$ by functoriality, hence can view $W_d\subseteq W(\hat G,\hat T)=N_{\hat G}/\hat T/\hat T$. Let $N_d$ be the inverse image of $W_d$ in $N_{\hat G}\hat T$. A bit work shows that both are in bijection with $\hat T(\mathbb{C})\times\{\mathrm{Fr}\}/N_d$.

Example 44 When $G$ is split, $S=T$, $\hat S=\hat T$ and $W_d=W(\hat G,T)=W(G,T)$. When $G=GL_n$, $\Phi(G)$ is simply the $GL_n(\mathbb{C})$-conjugacy classes of parameters $\phi: W_F\times SL_2(\mathbb{C})\rightarrow GL_n(\mathbb{C})$. The unramified parameters simply correspond to semisimple conjugacy classes in $GL_n(\mathbb{C})$, i.e., $S_n$-orbits of diagonal matrices. On the other hand, an unramified representation $\pi\in\Irr_{GL_n(F)}$ is a subquotient of a parabolic induction $\iota_B^G\chi_1 \otimes \cdots \otimes \chi_n$, where $\chi_i: F^\times\rightarrow \mathbb{C}^\times$ are unramified characters, determined up to $S_n$-conjugacy. The bijection is simply $\pi\mapsto\diag\{\chi_1(\varpi),\cdots,\chi_n(\varpi)\}$.

TopGlobal Langlands functoriality conjecture

Let $F$ be a number field and $G/F$ be a reductive group. Recall that for almost all places $v$ of $F$, the group $G_{F_v}$ is unramified. If $\pi$ is an irreducible admissible representation of $G(\mathbb{A}_F)$, then for almost all $v$, $\pi_v$ is an unramified $G_{F_v}$ representation. Choose for every place $v$ of $F$ an algebraic closure $\overline{F_v}$ and an embedding $\overline{F}\rightarrow\overline{F_v}$ extending $F\rightarrow F_v$. This induces an inclusion $\Gamma_{F_v}\hookrightarrow \Gamma_F$, hence a map ${}^{L}G_{F_v}\rightarrow {}^{L}G$.

Definition 89 For any irreducible admissible representation $\pi$ of $G(\mathbb{A}_F^\infty)$, the unramified local Langlands correspondence gives for almost all places $v$ a $\hat G(\mathbb{C})$-conjugacy classes $t_{\pi_v}\subseteq {}^{L}G_{F_v}\subseteq {}^{L}G_F$. We call the collection of elements $t_{\pi_v}$ defined for almost all $v$ the Satake parameters of $\pi$.
Definition 90 If $G,H$ are two reductive groups over $F$. An admissible homomorphism $\rho: {}^{L}G \rightarrow {}^{L}H$ is a continuous homomorphism satisfying
  1. The diagram $$\xymatrix{ {}^{L}G \ar[r] \ar[d] & {}^{L}H \ar[d]\\ \Gamma_F \ar[r]^\cong & \Gamma_F} $$ commutes.
  2. The underlying homomorphism $\hat G(\mathbb{C})\rightarrow \hat H(\mathbb{C})$ is induced by a homomorphism $\hat G\rightarrow\hat H$ of algebraic groups over $\mathbb{C}$.
Conjecture 3 (Global Langlands conjecture) Suppose $\rho: {}^{L}G \rightarrow {}^{L}H$ is an admissible homomorphism and $\pi$ is an automorphic representation of $G(\mathbb{A}_F)$. Assume $H $ is quasi-split. Then there exists an automorphic representation $\sigma$ of $H(\mathbb{A}_F)$ such that $t_{\sigma_v}=\rho(t_{\pi_v})$ for almost all places $v$.
Remark 80 There is no "global Langlands parameters": these should be homomorphisms from a conjectural Langlands group to ${}^{L}G$, which is elusive at the moment and one better not to think about them. The Weil group of a global field is of abelian nature but Galois representations arising from geometry are of highly non-abelian nature.

The are endless interesting examples of this conjecture by taking different $G$ and $H $.

Example 45 When $H $ is a quasi-split inner form of $G$ and $\rho: {}^{L}G \rightarrow {}^{L}H$ is the identity, the conjecture says that there is an automorphic representation $\sigma$ of $H(\mathbb{A}_F)$ such that $\pi_v\cong\sigma_v$.
Example 46 When $G=G_A$ (Example 13 ) and $H=GL_n$. This is the Jacquet-Langlands correspondence. You may not always go back from $H $ to $G$ due to the local obstruction.
Example 47 When $G=\{1\}$ and $H=GL_n$. A homomorphism $\tau: \Gamma_F\rightarrow GL_n(\mathbb{C})$ gives gives an admissible homomorphism ${}^{L}G\rightarrow {}^{L}H$. The conjecture says that there exists an automorphic representation $\pi$ of $GL_n(\mathbb{A}_F)$ such that $t_{\pi_v}$ is the conjugacy class of $\tau(\Frob_v)$. This is known as the strong Artin conjecture.


TopGlobal Langlands correspondence

References for this section:

  • Clozel and Milne, Ann Arbor volumes (the article by Clozel is especially relevant);
  • Buzzard and Gee, On the conjectural relations between automorphic representations and Galois representations (a more modern treatment).

There are Galois representations which do not correspond to automorphic representations (and vice versa). We have to make restriction on both sides in order to make sense of the global Langlands correspondence. This requires the notion of "algebraic" automorphic representations and "algebraic" Galois representations.

TopAlgebraic Galois representations

Let $F$ be a number field. Fix an algebraic closure $\overline{F}/F$. For a place $v$ of $F$, fix an algebraic closure $\overline{F_v}$ of $F_v$. Choose an embedding $\overline{F}\hookrightarrow \overline{F_v}$ extending the embedding $F\hookrightarrow F_v$. This induces an map $\Gamma_{F_v}\rightarrow \Gamma_F$, whose image is the decomposition group at $v$.

For $S$ a finite set of finite places of $F$, let $F^S\subseteq \overline{F}$ be the maximal unramified extension of $F$ away from $S$. Write $\Gamma_{F,S}=\Gal(F^S/F)$. For $v\not\in S$, the map $\Gamma_{F_v}\rightarrow \Gamma_F\rightarrow\Gamma_{F,S}$ factors through $\Gamma_{F_v}\rightarrow\hat{\mathbb{Z}}$. and we write $\Frob_v\in \Gamma_{F,S}$ for the image of the geometric Frobenius, i.e., the inverse the of the arithmetic Frobenius (which acts on the residue field by $x\mapsto x^q$).

Let $G/F$ be an reductive group. Let $\ell$ be a prime and $\overline{\mathbb{Q}_\ell}$ be an algebraic closure of $\mathbb{Q}_\ell$. Let $K/F$ be a finite Galois extension which splits $G$. We view $\hat G$ as the dual group defined over $\overline{\mathbb{Q}_\ell}$ and ${}^{L}G=\hat G\rtimes \Gal(K/F)$, a linear algebraic group over $\overline{\mathbb{Q}_\ell}$ with connected component $\hat G$. We endow ${}^{L}G$ with its natural $\ell$-adic topology induced from some embedding ${}^{L}G\subseteq GL_N(\overline{\mathbb{Q}_\ell})$ (it is not locally profinite because $\overline{\mathbb{Q}_\ell}$ is too big).

Definition 91 A continuous homomorphism $\rho: \Gamma_F\rightarrow {}^{L}G(\overline{\mathbb{Q}_\ell})$ is admissible if the composite map $\Gamma_F\rightarrow {}^{L}G(\overline{\mathbb{Q}_\ell})\rightarrow \Gal(K/F)$ is the natural projection.
Definition 92
  1. We say that a continuous homomorphism $\rho:\Gamma_F\rightarrow GL_n(\overline{\mathbb{Q}_\ell})$ is algebraic if
    1. there exists a finite set $S$ such that $\rho$ factors through $\Gamma_{F,S}$ (i.e., $\rho$ is unramified almost everywhere).
    2. For any place $v\mid\ell$ of $F$, the restriction $\rho|_{\Gamma_{F_v}}:\Gamma_{F_v}\rightarrow GL_n(\overline{\mathbb{Q}_\ell})$ is de Rham (the $v\mid \ell$-analogue of being "potentially semistable" at places $v\nmid\ell$).
  2. We say an admissible homomorphism $\rho: \Gamma_F\rightarrow {}^{L}G(\overline{\mathbb{Q}_\ell})$is algebraic if for any algebraic representation $R: {}^{L}G\rightarrow GL_N$, $R\circ\rho$ is algebraic. Equivalently, $R\circ\rho$ is algebraic for one faithfully representation $R$.
Example 48 Let $X$ be a smooth geometrically connected projective variety over $F$. The $\ell$-adic etale cohomology groups $H^*(X_{\overline{F}}, \overline{\mathbb{Q}_\ell})$ are finite dimensional $\overline{\mathbb{Q}_\ell}$-vector spaces on which $\Gamma_F$ acts. The associated Galois representations $\rho: \Gamma_F\rightarrow GL(H^*(X_{\overline{F}},\overline{\mathbb{Q}_\ell}))$ are algebraic. To prove that it factors through $\Gamma_{F,S}$, one can apply the proper smooth base change theorem after constructing a proper smooth model of $X$ away from $S$. To prove that it is de Rham, one needs Faltings' comparison theorem.
Example 49 When $X=E$ is the elliptic curve, $H^1(E_{\overline{F}}, \overline{\mathbb{Q}_\ell})$ is dual to $T_\ell(E)\otimes \overline{\mathbb{Q}_\ell}$. After choosing a basis, we obtain a 2-dimensional Galois representation $\rho: \Gamma_F\rightarrow GL_2(\overline{\mathbb{Q}_\ell})$. When $v\nmid\ell$ is a place of good reduction, $\rho$ is unramified at $v$ and $\tr\rho(\Frob_v)$ is an integer related the number of points of $E $ mod $\mathfrak{p}_v$.
Remark 81 There are only countably many Galois representations coming from algebraic geometry, but there are continuous deformation space of $\ell$-adic Galois representations. A heuristic calculation shows the "de Rham" condition cuts down the deformation space to zero dimension and there are only countably many Galois representations thus obtained. People tend to believe this subtle notion of being "de Rham" is the right notion to ensure the Galois representation comes from geometry (the Fontaine-Mazur conjecture) and should correspond to algebraic automorphic representations.

TopAlgebraic automorphic representations

Let $F$ be a number field and $G/F$ be a reductive group. Fix a place $v\mid\infty$ of $F$, induced by an embedding $F\subseteq \mathbb{C}$. Then $G(F_v)$ is a real Lie group. Let $\mathfrak{g}$ be the complexified Lie algebra. Choose $K\subseteq G(F_v)$ a maximal compact subgroup. Choose $T\subseteq G_\mathbb{C}$ a maximal torus and write $\mathfrak{t}=\Lie(T)$, $W=W(G_\mathbb{C},T_\mathbb{C})$.

Recall that we have a Harish-Chandra isomorphism $\mathbb{Z}(\mathfrak{g})\cong U(\mathfrak{t})^W$ (Theorem 24). Also recall that if $V$ is an irreducible admissible $(\mathfrak{g},K)$-module, then there exists $\lambda\in \mathfrak{t}^*$ such that the infinitesimal character $\chi_V$ of $V$ is equal to $\chi_\lambda$, obtained by $\mathbb{Z}(\mathfrak{g})\cong U(\mathfrak{t})^W\subseteq U(\mathfrak{t})\xrightarrow{\lambda} \mathbb{C}$. This determines $\lambda$ up to $W$-conjugacy. (Definition 73).

Notice $\mathfrak{t}^*$ has a natural integral lattice given by $X^\cdot(T)\hookrightarrow \mathfrak{t}^*$ given by $\alpha\mapsto d\alpha$.

Definition 93
  1. We say $V$ is $L $-algebraic if $\lambda_V$ lies in $X^\cdot(T)$.
  2. We say $V$ is $C $-algebraic if $\lambda_V+\rho\in X^\cdot(T)$, where $\rho$ is the half sum of the positive roots for some root basis. One can check this definition is independent of the choice of the root basis.
  3. If $\pi$ is an automorphic representation of $G(\mathbb{A}_F)$. We say $\pi$ is $L $-algebraic (resp. $C $-algebraic) if for any $v\mid\infty$, $\pi_v$ is $L $-algebraic (resp. $C $-algebraic).
Remark 82
  1. In general $\rho\in X^\cdot(T)\otimes \mathbb{Q}$ is not necessarily integral. But if $\rho\in X^\cdot(T)$ is integral (e.g, for $GL_n$, $n $ is odd), the two notions coincide.
  2. $C $ stands for Clozel, or cohomological. These automorphic representations were introduced by Clozel, which are supposed to contribute to the cohomology of locally symmetric spaces (e.g., Shimura varieties, or arithmetic hyperbolic 3-folds). But if $\pi$ is $C $-algebraic, this isn't always a Galois representation landing in ${}^{L}G(\overline{\mathbb{Q}_\ell})$. e.g., the automorphic representation attached to an elliptic curve $E/\mathbb{Q}$ descends to the group $G=PGL_2$, but there is no way to twist the associated Galois representation so that the image lies in ${}^{L}G=SL_2(\overline{\mathbb{Q}_\ell})$. The notion of $L $-algebraicity is a remedy of this drawback.
Example 50 Consider $G=GL_n$. If $\pi$ is an automorphic representation of $GL_n$, then for each $v\mid\infty$ we obtain $\chi_{\pi_v}\in X^\cdot(T)\otimes \mathbb{C}=\mathbb{Z}^n \otimes  \mathbb{C}$. If $\rho: \Gamma_F\rightarrow GL_n(\overline{\mathbb{Q}_\ell})$ is a Galois representation, then for each $v\mid\ell$ we obtain a Hodge-Tate-Sen weight lying in $X^\cdot(T) \otimes \overline{\mathbb{Q}_\ell}\cong \mathbb{Z}^n \otimes \overline{\mathbb{Q}_\ell}$. These weights all vary continuously. But if $\rho$ is de Rham, then the Hodge-Tate-Sen weights lie in $X^\cdot(T)$. The weights at $v\mid\infty$ can be read off from the Hodge structure of the corresponding motive. People guess these are related to the Hodge-Tate-Sen weights exactly when the de Rham condition is satisfied. There do exist Hodge-Tate but non de Rham Galois representations (e.g., coming from $\ell$-adic modular forms), but these don't come from geometry, see the end of the paper of Mazur-Wiles for an example.
Example 51 Fix $k\ge2$ and $N\ge1$. Assume given $f\in S_k(\Gamma_1(N),\mathbb{C})$ an eigenform for $T_p$, $p\nmid N$ with eigenvalue $a_p$. We associated $f$ an automorphic representation $\pi$ of $GL_2(\mathbb{A}_\mathbb{Q})$. For $p\nmid N$, $\pi_p=\iota_{B(\mathbb{Q}_p)}^{GL_2(\mathbb{Q}_p)}\chi_1 \otimes \chi_2$, where $\chi_1,\chi_2$ are unramified characters such that t$\chi_1(p)+\chi_2(p)=a_p/p^{1/2}$. Let $$|\cdot|:\mathbb{Q}^\times\backslash\mathbb{A}_\mathbb{Q}^\times\rightarrow \mathbb{C}^\times,\quad (x_p,x_\infty)\mapsto \prod_{p\le \infty} |x_p|_p.$$ Let $\pi_s=\pi \otimes |\det(g)|^s$ for any $s\in \mathbb{C}$. These $\pi_s$ are also cuspidal automorphic representations, which we can view as being associated to $f$ too. Sometimes it is convenient to normalize $\pi_s$ to be unitary, i.e. $\Re s=(k-2)/2$.

Notice $Z(\mathfrak{g})=\mathbb{C}[z,\Delta]\xrightarrow{\gamma} U(\mathfrak{t})^W$, where $z=\left(\begin{smallmatrix}1 & 0\\0&1\end{smallmatrix}\right)$ and $\gamma(z)=z$, $\gamma(\Delta)=(H^2-1)/2$. An algebra homomorphism $U(\mathfrak{t})^W\rightarrow \mathbb{C}$ is induced by an element of $X^\cdot(T)$ if and only if there exists $a,b\in \mathbb{Z}$ such that $z\mapsto a+b$, $H\mapsto a-b$, i.e., $\Delta\mapsto 1/2((a-b)^2+1)$. On the other hand, the $\pi_s$ has infinitesimal character $z\mapsto 2s+k-2$ and $\Delta\mapsto 1/2((k-1)^2-1)$.

Thus $\pi_s$ is $L $-algebraic if and only if $s\in 1/2+\mathbb{Z}$; $C $-algebraic if and only if $s\in \mathbb{Z}$. $\pi$ has a twist which is both $L $-algebraic and unitary if and only $k$ is odd. In particular, it explains the $k=2$ case in Remark 82.

Example 52 Let $F/\mathbb{Q}$ be a totally real field of degree $d$. Let $f$ be a cuspidal Hilbert modular form of weight $(k_1,\ldots,k_d)$, where $k_i\ge2$. One can associate $f$ a cuspidal automorphic representations $\pi$ which is defined up to a character twist. When does $\pi$ have an $L $-algebraic twist? Let $v_i$ be a real place of $F$, then $GL_2(F_v)=GL_2(\mathbb{R})$. The local theory at $v_i$ is the same for $F=\mathbb{Q}$. Suppose $\pi$ is $L $-algebraic, then $s_i\in 1/2+\mathbb{Z}$, $2s_i+k_i-2\in \mathbb{Z}$. Since $\pi$ is cuspidal, there exists $w\in \mathbb{R}$, such that $\pi \otimes |\cdot|^w$ is unitary (this is true for any cuspidal automorphic representations on general reductive groups). So $w=2s_i+k_i-2$ is independent of $i$. In particular, $k_i\mod{2}$ is independent of $i$. We conclude that one can lift $f$ to an $L $-algebraic $\pi$ only if the parity of $k_i$ is independent of $i$. In fact one can show this is also sufficient (see Clozel in Ann Arbor).

Can associate to each $\pi_{v_i}$ an $s_i\in \mathbb{C}$ such that $\pi_{v_i}$ has infinitesimal character.


TopGlobal Langlands correspondence

Fix a prime $\ell$. Choose an isomorphism $\iota: \overline{\mathbb{Q}_\ell}\cong \mathbb{C}$ (One expect everything to be defined algebraically, so this choice not essential. If we know everything is defined over $\overline{\mathbb{Q}}$, it is enough instead to fix something weaker: two embeddings $\overline{\mathbb{Q}}\hookrightarrow \mathbb{C}$ and $\overline{\mathbb{Q}}\hookrightarrow \overline{\mathbb{Q}_\ell}$) (see Remark 90). To state the general conjecture, we'd better fix such an isomorphism $\iota: \overline{\mathbb{Q}_\ell}\cong \mathbb{C}$.

Let $G/F$ be an reductive group over a number field $F$. Suppose $\pi$ is an automorphic representation of $G$.

Conjecture 4 Suppose $\pi$ is $L $-algebraic. Then there exists a finite set $T$ places of $F$ containing infinite places, places $v\mid\ell $ and the places at which $G_{F_v}$, $\pi_v$ are ramified, and an algebraic Galois representation $r_\iota(\pi): \Gamma_F\rightarrow {}^{L}G(\overline{\mathbb{Q}_\ell})$ satisfying:
  1. $r_\iota(\pi)$ is unramified outside $T$;
  2. For any $v\not\in T$, $r_\iota(\pi)(\Frob_v)\in \iota^{-1}t_{\pi_v}$ (Definition 89).
Remark 83 Buzzard-Gee pinned down the $L $-algebraicity condition in their recent paper. But this conjecture is not optimal: for example, the conditions do not determine $r_\iota(\pi)$ uniquely (up to $\hat G(\overline{\mathbb{Q}_\ell})$-conjugation), even in the case where $G$ is a torus!
Remark 84 If $G=GL_n$, these conditions do determine $r_\iota(\pi)$ uniquely up to semisimplification and $\hat G(\overline{\mathbb{Q}_\ell})$-conjugacy. The reason is that we have
  1. the Chebotarev density theorem that $\{\Frob_v\}_{v\in T}\subseteq \Gamma_{F,T}$ is a dense subset; and
  2. the representations of $GL_n$ are determined by their characters.
Remark 85 The choice of the isomorphism $\iota$ is important: different choices of $\iota$ will give different Galois representations associated to $\pi$.

In the case $G=GL_n$, we have the following more precise conjecture.

Conjecture 5 (Clozel+Fontaine-Mazur) Let $S$ be a finite set of places of $F$ containing the infinite places, places $v\mid\ell$. Then there exists a bijection between:
  1. irreducible algebraic representations $\rho: \Gamma_F\rightarrow GL_n(\overline{\mathbb{Q}_\ell})$ unramified outside $S$ up to isomorphism;
  2. cuspidal $L $-algebraic automorphic representations $\pi$ of $GL_n$ unramified outside $S$ satisfying that for $v\not\in S$, $r_\iota(\pi)(\Frob_v)\in \iota^{-1}t_{\pi_v}$.
Remark 86 Notice we do not specify the cuspidality condition for general $G$. The reason is that the notion of cuspidality does not generalize well: for example, there are cuspidal automorphic representations for unitary groups that correspond to reducible Galois representations (or mixed motives if you like). The Ramanujan conjecture for $GL_n$ says that $\pi$ is cuspidal then its local components $\pi_v$ are tempered (which reduces to the usual Ramanujan conjecture on the growth of Fourier coefficients of cusp forms when $n=2$, proved by Deligne). This conjecture is not true for general groups without appropriate modification.


If $v$ is a finite place of $F$, we denote the Artin map from local class field by $\Art_{F_v}: F_v^\times\rightarrow \Gamma_{F_v}^\mathrm{ab}$, normalized so that $\Art_{F_v}(\varpi_v)|_{F_v^\mathrm{ur}}=\Frob_v$, where $\Frob_v$ is the geometric Frobenius. We denote $\Art_F: \mathbb{A}_F^\times\rightarrow\Gamma_F^\mathrm{ab}$ for $\Art_F=\prod_v\Art_{F_v}$.

Remark 87 An automorphic representation of $GL_1(\mathbb{A}_F)=\mathbb{A}_F^\times$ is nothing but a Hecke character, i.e., a continuous character $\chi=\prod_v\chi_v: F^\times\backslash \mathbb{A}_F^\times\rightarrow \mathbb{C}^\times$. Notice is $\chi$ is $L $-algebraic if and only it is $C $-algebraic, since $GL_1$ has no roots.

When is a Hecke character $\chi$ $L $-algebraic? We need to look at the infinite components of $\chi$.

If $v$ is a real place, then $\chi_v:F_v^\times\cong\mathbb{R}^\times\rightarrow \mathbb{C}^\times$ is is given by $\chi_v(x)=|x|^s\sgn(x)^c$, where $s\in \mathbb{C}$, $c\in\{0,1\}$. The condition of $L $-algebraicity asks that the differential of $\chi_v$ agrees with the differential of an algebraic character of $GL_1$. That is to say, $s\in \mathbb{Z}$ (and no condition on $c$).

If $v$ is a complex place, then $\chi_v: F_v^\times\cong \mathbb{C}^\times\rightarrow \mathbb{C}^\times$. It has the form $\chi_v(z)=(z/|z|)^p|z|^s=z^{(s+p)/2}\bar z^{(s-p)/2}$, where $p\in \mathbb{Z}$, $s\in \mathbb{C}$ and the symbol $\bar z$ is defined formally so that $|z|=(z\bar z)^{1/2}$. We are supposed to think of $G_v=GL_1(F_v)$ as a real Lie group. Let $S=\Res_{\mathbb{C} /\mathbb{R}}GL_1$, it is a rank 2 torus over $\mathbb{R}$ whose functor points is $S(R)=(R \otimes_\mathbb{R}\mathbb{C})^\times$ for any $\mathbb{R}$-algebra $R$. In particular, $S(\mathbb{R})=\mathbb{C}^\times$. The $L $-algebraicity condition says that the differential of $\chi_v$ agrees with the differential of an algebraic character of $S$, i.e., $z\mapsto z^a\bar z^b$ for $a,b\in \mathbb{Z}$. That is to say, $s+p\in 2 \mathbb{Z}$.

These $L $-algebraic Hecke characters are exactly the character of type $A_0$ already introduced by Weil.

Remark 88 The complex characters of $\Gamma_F$ are of finite order, which corresponds to Hecke characters with trivial infinite components. But there are many interesting examples of Hecke characters with nontrivial infinite components. For example, the $\ell$-adic Tate module of an elliptic curve over $\mathbb{Q}$ with CM by an imaginary quadratic field $K$ gives a Hecke character on $K^\times\backslash\mathbb{A}_K^\times$ of infinite type $(a,b)=(-1,0)$.

Here is a compact way of describing an $L $-algebraic character $\chi$: there exits integers $n_\tau$ indexed by embeddings $\tau: F\rightarrow \mathbb{C}$ such that $\chi(x)=\prod_{\tau: F\rightarrow \mathbb{C}}x_\tau^{n_\tau}$, where $x=(x_v)\in ((F \otimes_\mathbb{Q}\mathbb{R})^\times)^0\subseteq \mathbb{A}_F^\times$ and $x_\tau=\tau(x_v)$, where $v$ is the place of $F$ induced by $\tau: F\rightarrow F_v\rightarrow \mathbb{C}$.

The following theorem verifies Conjecture 5 for $G=GL_1$.

Theorem 37 Fix a prime $\ell$ and $\iota:\overline{\mathbb{Q}_\ell} \cong \mathbb{C}$. Let $\chi: F^\times\backslash\mathbb{A}_F^\times\rightarrow \mathbb{C}^\times$ be an $L $-algebraic character. Then there exists a unique representation $r_\iota(\chi):\Gamma_F\rightarrow \overline{\mathbb{Q}_\ell}^\times$ satisfying the condition of Conjecture 5. Explicitly, it is given by $$\iota((r_\iota(\chi)\circ\Art_F(x)\prod_{\tau: F\hookrightarrow \overline{\mathbb{Q}_\ell}} x_\tau^{-n_{\iota\tau}})=\chi(x)\prod_{\tau: F\rightarrow \mathbb{C}}x_\tau^{-n_\gamma}, x\in\mathbb{A}_F^\times.$$ Conversely, every algebraic representation $\rho: \Gamma_F\rightarrow \overline{\mathbb{Q}_\ell}^\times$ arises in this way from a unique $L $-algebraic character $\chi$.
Remark 89 The integers $n_\tau$ are the Hodge-Tate weights of $r_\iota(\chi)$ (up to sign).
Remark 90 One can show that $\chi(x)\prod x_\tau^{-n_\tau}$ really takes value in $\overline{\mathbb{Q}}^\times\subseteq \mathbb{C}^\times$. In other words, one pulls out the contribution of $\chi$ at infinite places so that it is valued in $\overline{\mathbb{Q}}$, hence can be viewed as valued in $\overline{\mathbb{Q}_\ell}$. One then puts back the contribution to the $\ell$-adic places and obtain the above formula.
Remark 91 To prove the theorem, one uses global class field theory and needs to understand the de Rham condition for $GL_1$. It turns out to be quite simple in this case: de Rham is the same as Hodge-Tate, which is the same as "locally algebraic". See, Serre, abelian $\ell$-adic representations.


We state partial results (automorphic to Galois) toward Conjecture 5 for $GL_n$ over CM fields.

Definition 94 A number field $E $ is called CM if there exists $c\in \Aut(E)$ such that $\tau(x^c)=\overline{\tau(x)}$, for any embedding $\tau: E\rightarrow \mathbb{C}$. Let $F=E^{c=1}$, then there are only two cases:
  1. $E=F$ is totally real;
  2. $F$ is totally real and $E/F$ is a totally complex quadratic extension.
Theorem 38 Let $E $ be a CM field and $\pi$ be a cuspidal $L $-algebraic automorphic representation of $GL_n(\mathbb{A}_E)$. Suppose
  1. If $E $ is real, then $\pi\cong\pi^\vee$.
  2. If $E $ is complex, then $\pi^c\cong\pi^\vee$, where $c$ acts by its action on $GL_n(\mathbb{A}_E)$.
  3. $\pi$ is regular, i.e, for any $\tau: E\hookrightarrow \mathbb{C}$, the element $\lambda_\tau\in X^\cdot(\hat T)$ associated to the infinitesimal characters, is regular.

Then for every $\ell$ and $\iota: \overline{\mathbb{Q}_\ell} \cong \mathbb{C}$, there exists an algebraic Galois representation $r_\iota(\pi):\Gamma_E\rightarrow GL_n(\overline{\mathbb{Q}_\ell})$ such that $r_\iota(\pi)(\Frob_v)_\mathrm{ss}\in \iota^{-1}t_{\pi_v}$, when $\pi_v$ is unramified.

Remark 92 It follows from the Tate conjecture that $r_\iota(\pi)(\Frob_v)$ is semisimple, but we don't know that yet.
Remark 93 The theorem is due to the work of many people. Most of these $r_\iota(\pi)$ can be found in the cohomology of Shimura varieties (thanks to the regularity condition); some of them need $\ell$-adic interpolation and more difficult techniques. See Chenevier-Harris in Cambridge Math. J. for a final version.
Remark 94 We actually know more: the theorem holds without hypothesis a), b) (Harris-Lan-Taylor-Thorne), but we don't yet know the associated Galois representations is de Rham at $v\mid \ell$ (since they are constructed using $\ell$-adic interpolation). Scholze has given another proof which gives much more.
Remark 95 We also know some cases when $\pi$ is not regular (Goldring) and in some cases when $G$ is a classical group other than $GL_n$, but these results are far from complete. There are no theorems in good generality over non CM fields.
Remark 96 Gross constructed an automorphic representation of $G_2$ that can be shown to be $L $-algebraic, can anyone construct the associated Galois representation valued in $G_2$?

TopLocal-global compatibility

Now suppose $F$ is any number field and $\pi$ is a cuspidal $L $-algebraic automorphic representation of $GL_n(\mathbb{A}_F)$. Suppose $\rho=r_\iota(\pi)$ is known to exist. The local-global compatibility should give $\rho|_{G_{F_v}}=\rec_{F_v}(\pi_v)$ at ramified places, where $\rec_{F_v}$ is the local Langlands correspondence. When $v\nmid \ell$, one can say exactly what $\rho|_{G_{F_v}}$ should be using Grothendieck's $\ell$-adic monodromy theorem (see Tate, Number theoretic background in Corvallis II).

Back to the situation in Theorem 38, the local-global compatibility is known to hold due to Taylor-Yoshida for the case a) and Caraiani for the case b).

We now state a special case we shall need for the application of Chenevier-Clozel on number fields with limited ramification.

Theorem 39 Fix a finite place $v_0\nmid \ell$ of $E $. Suppose $\pi_v$ is unramified if $v\ne v_0$ and $\pi_{v_0}$ is supercuspidal. Then $\rec_{E_v}(\pi_{v_0}): W_{E_{v_0}}\rightarrow GL_n(\overline{\mathbb{Q}_\ell})$ is irreducible. The representation $\rho=r_\iota(\pi)$ is unramified at all $v\nmid \ell v_0$, and $\rho|_{W_{E_{v_0}}}\cong\rec_{E_{v_0}}(\pi_{v_0}) \otimes_{\iota^{-1},\mathbb{C}} \overline{\mathbb{Q}_\ell}$.


References for this section:

  • Chenevier, Number fields with given ramification (Compositio)
  • Chenevier-Clozel, Corps de nombres peu ramifies (JAMS)

TopNumber fields with given ramification

The remaining of this course will be devoted to the application of Chenevier-Clozel on number fields with prescribed ramification, as promised in the first class. To put things in context, we first recall the following classical result.

Theorem 40 Let $p$ a prime and $K/\mathbb{Q}_p$ be a finite extension. Then there exists an extension $L/\mathbb{Q}$ such that $[L:\mathbb{Q}]=[K:\mathbb{Q}_p]$ and $L_v\cong K$ as $\mathbb{Q}_p$-algebras where $v$ is the unique place of $L $ over $p$.
Proof Choose $\alpha\in K$ be a primitive element (i.e., $K=\mathbb{Q}_p(\alpha)$). Let $f_p(x)=\sum a_ix^i\in \mathbb{Q}_p[x]$ be its (monic) minimal polynomial. One can choose $f(x)=\sum b_ix^i\in \mathbb{Q}[x]$ such that $|a_i-b_i|_p<\varepsilon$ for any $i$ and $\varepsilon>0$. By Krasner's lemma, if $\varepsilon$ is small enough, then $f(x)$ has a root in $K$ and $K=\mathbb{Q}_p(\alpha)$. We simply set $L=\mathbb{Q}[x]/f(x)$. Then $L \otimes_\mathbb{Q} \mathbb{Q}_p=\mathbb{Q}_p[x]/f(x)\cong K$.
Corollary 7 Choose algebraic closures $\overline{\mathbb{Q}}$ of $\mathbb{Q}$ and $\overline{\mathbb{Q}_p}$ of $\mathbb{Q}_p$ and an embedding $j: \overline{\mathbb{Q}}\hookrightarrow \overline{\mathbb{Q}_p}$. Then the natural map $j^*: \Gal(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)\rightarrow \Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ is injective.
Remark 97 The proof of this theorem does not give control of the ramification away from $p$. In general, the Artin-Whaples approximation theorem controls the polynomial $f$ at finitely many places.

We would like to control the ramification at almost all primes. Consider a finite set $S$ of primes and a prime $p\in S$. Let $\mathbb{Q}_S$ the maximal unramified subfield of $\overline{\mathbb{Q}}$ unramified outside $S$.

Definition 95 We write $P_{S,p}$ for the property that the natural map $\Gal(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)\rightarrow\Gal(\mathbb{Q}_S/\mathbb{Q})$ is injective. Equivalently, if $K/\mathbb{Q}_p$ is a finite extension, then there exists a number field $L/\mathbb{Q}$ unramified outside $S$, a place $v$ of $L $ above $p$ and an embedding $K\hookrightarrow L_v$ of $\mathbb{Q}_p$-algebras.

Chenevier-Clozel proved that this property is true as long as $S$ contains at least two primes.

Theorem 41 (Chenevier-Clozel) $P_{\{p,\ell\}, p}$ holds if $p$ and $\ell$ are distinct primes.

More generally,

Definition 96 Let $E $ is any number field, $S$ be a finite set of finite places of $E$ and $v\in S$. Write $P_{E,S,v}$ similarly for the property that $\Gal(\overline{E_v}/E_v)\rightarrow\Gal(E_S/E)$ is injective.

We are going to construct interesting automorphic representations with prescribed level, whose associated Galois representation helps us to attack this algebraic number theoretic problem $P_{E,s,v}$.

Lemma 12 Let $m,r>1$ be integers and $p$ be a prime. Then the set of primes $\ell$ such that the order of $m$ in $\mathbb{F}_\ell^\times$ is divisible by $p^r$ is infinite.
Proof Since $r $ is arbitrary, it suffices to show that this set is nonempty. Let $$t=(m^{p^r}-1)/(m^{p^{r-1}}-1)=(m^{p^{r-1}}-1)^{p-1}+p(m^{p^{r-1}}-1)^{p-2}+\cdots +p.$$ Let $\ell$ be a prime dividing $t$. If $\ell\nmid (m^{p^{r-1}}-1)$, then we are done. Otherwise, the above expression for $t$ gives $\ell\mid p$. There are two cases.
  1. If $p>2$, then $p^2\mid (m^{p^{r-1}}-1)^{p-1}$, hence $p^2\nmid t$. But $t>p$, so we can find another prime $\ell\mid p$, a contradiction.
  2. If $p=2$, $t=(m^{2^{r-1}}-1)+2$, then $4\nmid t$ and the same argument gives a contradiction.
Lemma 13 Let $M$ be a finite extension of $\mathbb{Q}_p$. Suppose $\sigma\in \Gamma_M$ acts trivially on $I_M^t$ (the tame quotient of the inertia subgroup $I_M$) by conjugation. Then $\sigma\in I_M$.
Proof If $\sigma\in \Gamma_M$, then by Kummer theory, its action on $I_M^t$ is the multiplication by $q^{\mu(\sigma)}\in\prod_{\ell\ne p}\mathbb{Z}_\ell^\times$, where $\mu: \Gamma_M\rightarrow \hat{\mathbb{Z}}$ is the unramified quotient and $q$ is the size of the residue field of $M$. If this action is trivial, the image of $q^{\mu(\sigma)}$ in $\mathbb{F}_\ell ^\times$ is trivial for all primes $\ell$. By the previous lemma applying to $m=q$, we know that $\mu(\sigma)=0\in \hat{\mathbb{Z}}/ n \hat{\mathbb{Z}}$, where $n $ is any prime power. Hence $\mu(\sigma)=0\in \hat{\mathbb{Z}}$ and $\sigma\in I_M$.

We can rephrase this lemma in the language of Galois theory.

Corollary 8 Let $M$ be a finite extension of $\mathbb{Q}_p$. Let $L/M $ be a (possibly infinite) Galois extension. If $L\cdot M^\mathrm{ur}=\overline{\mathbb{Q}_p}$, then $L=\overline{\mathbb{Q}_p}$.
Proof Let $H=\Gal(\overline{\mathbb{Q}_p} /L)$ and $I_M=\Gal(\overline{\mathbb{Q}_p}/M^\mathrm{ur})$. They are closed normal subgroups of $\Gamma_M$. The assumption shows that $H\cap I_M=\{1\}$. Hence $H I_M=H\times I_M\subseteq \Gamma_M$. In particular, $H $ commutes with $I_M$ , hence by the previous lemma, $H\subseteq I_M$, which is impossible unless $H=\{1\}$.
Lemma 14 Let $E $ be a number field and $S$ be a finite set of finite places of $E $ , $v\in S$. Choose $\ell$ a prime, an isomorphism $\iota: \overline{\mathbb{Q}_\ell} \cong \mathbb{C}$ and an embedding $j: \overline{E}\rightarrow \overline{E_v}$ extending the canonical map $E\rightarrow E_v$. Suppose for every irreducible continuous representation $\rho: \Gal(\overline{E_v}/E_v)\rightarrow GL_n(\mathbb{C})$ there exists a continuous representation $R: \Gal(E_S/E)\rightarrow GL_n(\overline{\mathbb{Q}_\ell})$ such that $$R|_{I_{E_v}}\cong (\rho \otimes_{\mathbb{C},\iota^{-1}} \overline{\mathbb{Q}_\ell})|_{I_{E_v}}.$$ Then the property $P_{E,S,v}$ holds.
Remark 98 The image of $\rho$ is finite since $GL_n(\mathbb{C})$ has no small subgroup.
Proof By the previous corollary, it is enough to show that $j(E_S)E_v^\mathrm{ur}=\overline{E_v}$ (so the completion of $j(E_S)$ is $\overline{E_v}$). Fix $K/E_v$ a finite Galois extension inside $\overline{E_v}$, we need to show that $K\subseteq j(E_S)E_v^\mathrm{ur}$. Let $T$ be the regular representation of $\Gal(K/E_v)$. Applying the hypothesis of the lemma to each irreducible representation $\tau$ (of $\Gal(K/E_v)$, we obtain a representation $R_\tau:\Gal(E_S/E)\rightarrow GL_n(\overline{\mathbb{Q}_\ell})$ such that $R_\tau|_{I_{E_v}}\cong\tau|_{I_{E_v}}$. Let $R=\oplus_\tau R_\tau^{\dim \tau}$. Then $R|_{I_{E_v}}\cong T|_{I_{E_v}}$. Since the regular representation is faithful, by Galois theory, we obtain that $K\subseteq j(E_S)E_v^\mathrm{ur}$: indeed, we have $\overline{E_v}^{\ker R}\subseteq j(E_S)E_v\subseteq \overline{E_v}$ and $\overline{E_v}^{\ker R|_{I_{E_v}}}\subseteq j(E_S) E_v^\mathrm{ur}$, by construction $\overline{E_v}^{\ker R|_{I_{E_v}}}=K\cdot E_v^\mathrm{ur}$.
Corollary 9 Let $E $ be totally complex CM field with its maximally totally real subfield $F\subseteq E$. Let $c\in \Gal(E/F)$ be the unique nontrivial element. Fix a prime $\ell$, a finite place $w_0$ of $E $ not dividing $\ell$ and a finite set $S$ of finite places of $E $ such that the places $w\mid \ell$ and $w_0$ are contained in $S$. Suppose that for all integers $n\ge1$ and for every supercuspidal representation $\pi_0$ of $GL_n(\pi_{w_0})$, there exists a cuspidal conjugate self dual regular $C $-algebraic automorphic representation $\Pi$ such that
  1. For any $w\not\in S$ of $E $, $\Pi_w$ is unramified;
  2. there exists an unramified character $\psi: E_{w_0}^\times\rightarrow \mathbb{C}^\times$ such that $\Pi_{w_0}\cong\pi_{w_0}\otimes(\psi\circ \det)$.

Then $P_{E,S,w_0}$ holds.

Remark 99 In other words, constructing enough automorphic representations satisfying the local properties will allow us to use the associated Galois representation to construct the required number fields.
Proof We will show that the conditions in the previous lemma hold. Fix $\iota:\overline{\mathbb{Q}_\ell}\cong \mathbb{C}$ and a continuous irreducible representation $\rho:\Gamma_{E_{w_0}}\rightarrow GL_n(\mathbb{C})$. We would like to realize it globally. Let $\pi_0=\rec^{-1}_{E_{w_0}}(\rho|_{W_{E_{w_0}}})$ be a supercuspidal representation of $GL_n(E_{w_0})$ (Example 22). Let $\Pi$ be an automorphic representation of $GL_n(\mathbb{A}_E)$ satisfying the assumption. Then $\Pi_1=\Pi \otimes |\cdot|^{(1-n)/2}$ is a cuspidal regular $L $-algebraic automorphic representation of $GL_n(\mathbb{A}_E)$ and has associated Galois representation (Theorem 38) $r_{\iota}(\Pi_1):\Gamma_E\rightarrow GL_n(\overline{\mathbb{Q}_\ell})$ satisfying
  1. $r_\iota(\Pi_1)$ is unramified at all places $w\not \in S$ of $E $ as $\Pi_{1,w}$ is unramified and $w\nmid \ell$.
  2. $r_\iota(\Pi_1)|_{I_{E_{w_0}}} \otimes_{\overline{\mathbb{Q}_\ell},\iota}\mathbb{C} \cong \rho|_{I_{E_{w_0}}}$. This follows from the local-global compatibility and the compatibility of local Langlands correspondence with twisting by characters. Notice $\psi|\cdot|^{(1-n)/2}$ is unramified so the twisting disappears when restricting to the inertia subgroup.

We won't prove the full $P_{\mathbb{Q},\{p,\ell\},p}$. Instead, we will prove an earlier result of Chenevier:

Theorem 42
  1. Let $E $ be a totally complex CM field and let $w_0$ be a place of $E $ split over $F$. Let $S=\{w\mid \ell\}\cup \{w_0,w_0^c\}$ and assume $w_0\nmid \ell$. Then $P_{E,S,w_0}$ holds.
  2. Let $p$ be a prime and $N $ be an integer such that $-N$ is the discriminant of a quadratic imaginary field in which $p$ splits. Let $S$ be the set of places dividing $pN$. Then $P_{\mathbb{Q}, S,p}$ holds.
Remark 100
  1. Part b) follows from a) by taking $E $ be the imaginary quadratic field in which $p$ splits.
  2. Examples of $(p,N)$ satisfying this condition: $(p,N)\in\{(2,7),(2,15),(3,8),(3,20),(3,11)\}$. We deduce that, for example, $\mathbb{Q}_{\{2,7\}}$ is dense in $\overline{\mathbb{Q}_2}$.


TopBase change from unitary groups


  • Mok, Endoscopic classification of representations of quasi-split unitary groups;
  • Clozel et. al., On the stabilization of the trace formula (Paris book project)

In view of Corollary 9, we would like to construct automorphic representations with prescribed local component. In some sense the only way of doing this is to use the trace formula. The problem is that, unlike the supercuspidal representations at finite places, the $C $-algebraic representations of $GL_n(\mathbb{C})$ are not isolated in the unitary dual of $GL_n(\mathbb{C})$ and hence is not easy to pick out by choosing suitable test functions. But they are isolated in the subset of conjugate self-dual representations of $GL_n(\mathbb{C})$. So one can use the twisted trace formula which picks out only conjugate self-dual ones. This is what Clozel used in proving $P_{\mathbb{Q},\{p,\ell\},p}$. We will take a different approach using functoriality (base change from a unitary group).

Let $F$ be a number field and $E/F$ be a quadratic extension with the nontrivial element $c\in\Gal(E/F)$. Let $G$ be a unitary group associated to a non-degenerate Hermitian form $\langle,\rangle$ on $E^n$. Write $\langle v,w\rangle= {}^tv^cJw$, $J\in M_n(E)$. So $G$ is the reductive group over $F$ with functor of points $G(R)=\{g\in GL_n(E \otimes_FR): {}^tg^cJg=J\}$. It is an outer form of $G$, split over $E $.

If $w$ is a place of $E $ above a place $v$ of $F$. There are two cases:

  1. $v$ is split in $E $, then $E \otimes_F F_v=E_w\times E_{w^c}$. Hence $G(F_v)=\{(g,h)\}\in GL_n(E_w)\times GL_n(E_{w^c}): {}^tg^cJh=J\}$. Projecting to the first factor shows this is isomorphic to $GL_n(E_w)$. Namely, $G_{F_v}\cong GL_{n,{E_w}}$ (though this isomorphism depends on a choice of the place $w$).
  2. $v$ is inert or ramified in $E $, then $G_{F_v}$ is a "true unitary group", which becomes split only after extension of scalars to $E_w$.
Remark 101 The $L $-group is ${}^{L}G=GL_n(\mathbb{C})\rtimes \Gal(E/F)$: $c$ acts on $GL_n(\mathbb{C})$ via $g\mapsto \alpha(g)=\Psi_n {}^t\bar g^{-1}\Psi_n^{-1}$, where $\Psi_n=\mathrm{antidiag}\{1,-1,\ldots,(-1)^{n-1}\}$. Notice $\alpha$ is the unique nontrivial automorphism of $GL_n(\mathbb{C})$ preserving the standard pinning (c.f., Example 13).
Lemma 15 Let $H=\Res_{E/F}GL_{n,E}$.
  1. $H $ is a form of $GL_n\times GL_n$.
  2. ${}^{L}H=(GL_n(\mathbb{C})\times GL_n(\mathbb{C}))\rtimes \Gal(E/F)$, where $c(g,h)c^{-1}=(h,g)$.
  3. If $w$ is a place of $E $ above $v$ of $F$, we have an injection between Langlands parameters $\Phi(G_{F_v})\hookrightarrow \Phi(H_{F_v})$ and a bijection $\Phi(H_{F_v})\cong\Phi(GL_{n,E_w})$.
Proof (See Mok for details) Notice for any $E $-algebra $R$, $$H(R)=GL_n((E\times E) \otimes_E R)=GL_n(E \otimes_ER)\times GL_n(E \otimes_{E,c}R).$$ So $H_E\cong GL_n\times GL_n$. One can then figure out the ${}^{L}H$. To get a map $\Phi(G_{F_v})\rightarrow \Phi(H_{F_v})$, we write down an admissible homomorphism between the $L $-groups: $f:{}^{L}G\rightarrow {}^{L}H$, given by $f(g\times 1)=(g,{}^t\bar g^{-1})\times 1$, $f(1\times c)=(\Phi_n,\Phi_n^{-1})\times c$ (one can check that it is admissible). To get a map $\Phi(H_{F_v})\rightarrow \Phi(GL_{n,F_v})$, starting with the parameter $\phi: W_{F_v}\times SL_2(\mathbb{C})\rightarrow {}^{L}H$. Then $$\phi|_{W_{E_w}\times SL_2(\mathbb{C})}: W_{E_w}\times SL_2(\mathbb{C})\rightarrow GL_n(\mathbb{C})\times GL_n(\mathbb{C})$$ is a pair of parameter $\phi_1,\phi_2: W_{E_w}\times SL_2(\mathbb{C})\rightarrow GL_n(\mathbb{C})$. The map is simply $\phi\mapsto\phi_1$. See Mok for the proof of the injectivity (resp. bijectivity) between the Langlands parameters.
Remark 102 The bijection is a special case of what Langlands calls "Shapiro's lemma for $L $-groups".

It follows that we have an injection $\Phi(G_{F_v})\hookrightarrow \Phi(GL_{n,E_w})$. Since the local Langlands correspondence for $GL_{n,E_w}$ is bijective, if we also know the local Langlands correspondence for $G_{F_v}$, then we obtain a map $\Irr(G(F_v))\rightarrow \Irr(GL_n(E_w))$. This map is called the local base change when it exists. The following cases are known to exist:

  1. When $v$ is split in $E $, we had an isomorphism $\iota_w:G_{F_v}\cong GL_{n,E_w}$ and this map is simply $\sigma_v\in\Irr(G(F_v))\mapsto\sigma_v\circ\iota_w^{-1}\in\Irr(GL_n(E_w))$.
  2. When $v$ is inert in $E $ and $\Phi(G_{F_v})$ is unramified, this map is explicitly described by Minguez in the Paris book project.
  3. When $v$ is archimedean.

The functoriality in this special case says the following:

Conjecture 6 (Functoriality for $f:{}^{L}G\rightarrow {}^{L}H$) Let $\sigma$ be an automorphism representation of $G(\mathbb{A}_F)$. Then there exists an automorphic representation $\Pi$ of $GL_n(\mathbb{A}_E)$ satisfying: for $v$ in the previous above cases, $\Pi_w$ is given by the local base change applied to $\sigma_v$.
Remark 103 $\Pi$ obtained this way is called weak base change, since it is unspecified at finitely many places. We say $\Pi$ is the strong base change if it is specified at all places.
Lemma 16 Suppose $\sigma$ is an automorphic representation of $G(\mathbb{A}_F)$ satisfying
  1. There exists a place $w_0$ of $E $ split over $v_0$ of $F$ such that $\sigma_{v_0}$ is supercuspidal.
  2. Base change $\Pi$ of $\sigma$ exists.

Then $\Pi$ is cuspidal and $\Pi^c\cong\Pi^\vee$ (conjugate self dual).

Proof By the theory of Eisenstein series, if $\Pi$ is not cuspidal, then for any place $w$ of $E $, $\Pi_w$ is a subquotient of a parabolic induction from a proper Levi subgroup (see Langlands supplement to Corvallis I). By the assumption, $\Pi_{w_0}$ is supercuspidal, which is not of the form, hence $\Pi$ itself is cuspidal. In particular, $\Pi^c$ and $\Pi^\vee$ are also cuspidal. Now by strong multiplicity one for cuspidal automorphic representations of $GL_n(\mathbb{A}_E)$, one only needs to check that $\Pi^c$ and $\Pi^\vee$ are isomorphic for almost all places $w$. This can be checked by the construction of the base change. For example, in the second case ($v$ is inert and $\sigma_v$ is unramified), then $\sigma_v$ has a parameter $\phi: W_{F_v}\rightarrow GL_n(\mathbb{C})\rtimes \Gal(E_w/F_v)$. The parameter for $\Pi_w$ is $\phi|_{W_{E_w}}: W_{E_w}\rightarrow GL_n(\mathbb{C})$, so the parameter for $(\Pi^c)_w$ is $(\phi|_{W_{E_w}})^c$ and the parameter for $\Pi_w^\vee$ is ${}^t(\phi|_{W_{E_w}})^{-1}$ (by the compatibility of the local Langlands correspondence for $GL_{n,E_w}$ with the passage to the contragradient). But $(\phi|_{W_{E_w}})^c=\phi(\tilde c)\phi|_{W_E}\phi(\tilde c)^{-1}$, where $\tilde c\in W_{F_v}$ is a lift of $c$ and $\phi(\tilde c)$ is of the form $g\times c$. A calculation shows that these two parameters agrees up to conjugation. (c.f., Remark 101).

The existence of the base change is the content of the next theorem.

Theorem 43 Let $E $ be a totally complex CM field and $F=E^{c=1}$. Let $G$ be a quasi-split unitary group in $n $ variables with respect to $E/F$. Let $\sigma$ be an automorphic representation of $G(\mathbb{A}_F)$ satisfying:
  1. There exists a place $v_0$ of $F$ split in $E $ such that $\sigma_{v_0}$ is supercuspidal.
  2. For all $v\mid \infty$ of $F$, $\sigma_v$ is square-integrable.

Then the base change $\Pi$ of $\sigma$ exists and is a cuspidal, conjugate self-dual, and regular $C $-algebraic automorphic representation of $GL_n(\mathbb{A}_E)$.

Remark 104 Base change is known to exist in much more generality: one does need the local assumptions and $E $ is not necessarily required to be a CM field.
Proof For the existence of $\Pi$, see Mok. We already know that $\Pi$ is cuspidal and conjugate self-dual. It remains to check the regularity and $C $-algebraicity at $w\mid \infty$ of $E $. The square-integrable representations of $G(F_v)$, $v\mid \infty$ is parametrized by their infinitesimal characters which came from regular elements $\lambda\in X^\cdot(T_\mathbb{C})$. If a $G(\mathbb{R})$ representations $\tau$ is square-integrable, then $\tau$ is regular and $C $-algebraic (Theorem 26). One then deduces the properties of the base change of $\tau$ by calculating the infinitesimal character under base change.

To summarize, to prove Theorem 42,we are reduced to, by the previous theorem and Corollary 9, the following theorem (which gives more than we needed as input for Corollary 9).

Theorem 44 Let $F$ be a number field and $G/F$ be a reductive group. Suppose that for any $v\mid\infty$ of $F$, $G(F_v)$ has compact center and $G(F_v)$ has square-integrable representations (this implies that $F$ is totally real). Fix a finite place $v_0$ of $F$ and a supercuspidal representation $\pi_0$ of $G(F_{v_0})$ and an open compact subgroup $K^{v_0}\subseteq G(\mathbb{A}_F^{\infty,v_0})$. Then there exists an automorphic representation $\sigma$ of $G(\mathbb{A}_F)$ satisfying
  1. For any $v\mid \infty$, $\sigma_v$ is square-integrable;
  2. There exists an unramified character $\psi: G(F_{v_0})\rightarrow \mathbb{C}^\times$ such that $\sigma_{v_0}\cong\pi_0 \otimes \psi$.
  3. $\sigma^{K^{v_0}}\ne0$.

In the remaining of this course, we will deduce this theorem from Arthur's simple trace formula.


TopApplication of the trace formula for compact quotients

To make life a bit easier, we are going to prove this theorem in the case where $F=\mathbb{Q}$, $v_0=p$ and $G$ has trivial center to avoid minor technicality. We will first focus on the case where $G(\mathbb{R})$ is compact and treat the general case next time.

Since $G(\mathbb{R})$ is compact, $G$ is anisotropic and we can use the trace formula for the compact quotient $G(\mathbb{Q})\backslash G(\mathbb{A})$ (Theorem 29).

Remark 105 Notice $G_\gamma$ has reductive connected component since every element $g\in G(\mathbb{Q})$ is semisimple ($G$ is anisotropic).
Remark 106 Let $C\subseteq G(\mathbb{A})$ be a compact subset. Then we can find a finite set $S_c\subseteq \{\gamma\}$ such that for $\phi\in C_c^\infty(G(\mathbb{A}))$ with $\supp(\phi)\subseteq C$, we have the orbital integral $\mathcal{O}_\gamma(\phi)\ne0$ only if $\gamma\in S_c$ by Arthur's reduction theory (Remark 73).

We will choose $\phi$ of the form $$\phi=\phi^{p,\infty} \otimes \phi_p \otimes \phi_\infty.$$ where $\phi^{p,\infty}\in C_c^\infty(G(\mathbb{A}^{p,\infty}))$, $\phi_p\in C_c^\infty(G(\mathbb{Q}_p))$, $\phi_\infty\in C_c^\infty(G(\mathbb{R}))$. We pick

  1. $\phi^{p,\infty}$ the characteristic function of $K^p$;
  2. $\phi_p$ the matrix coefficient of $\pi_0^\vee$ (it is compactly supported as $\pi_0$ is supercuspidal and $G$ has trivial center).
  3. This is the most interesting one: $\phi_\infty=\phi_\infty^\lambda$ is allowed to vary depending on a parameter $\lambda$. Fix $T\subseteq G_\mathbb{R}$ a maximal torus, $W=W(G_\mathbb{C},T_\mathbb{C})$, $\Phi=X^\cdot(T_\mathbb{C})$ set of roots of $G_\mathbb{C}$. $S\subseteq \Phi$ a root basis. $\Phi^+$ the set of $S$-positive roots. We parametrize the irreducible representations of $G(\mathbb{R})$ with the $S$-dominant weights $\lambda\in X^\cdot(T_\mathbb{C})$ (Theorem 21). Write $(\pi_\lambda,V_\lambda)$ the corresponding highest weight representation for $\lambda$. We take $\phi_\infty^\lambda(\gamma)=\tr\pi_\lambda(\gamma)$.

We need to know about these functions in order to proceed. The key input is the Weyl character formula, which we now review.

Definition 97 Let $x_1,\ldots,x_n$ be a basis of the free $\mathbb{Z}$-module $X^\cdot(T_\mathbb{C})\cong \Hom(T(\mathbb{R}),\mathbb{C}^\times).$ We say a function $Q:X^\cdot(T_\mathbb{C})\rightarrow \mathbb{Z}$ is polynomial if it is given by a polynomial in (the dual basis of) $x_1,\ldots,x_n$. We say a function $f: T(\mathbb{R})\rightarrow \mathbb{C}$ is rational if it is given by a rational function of $x_1,\dots,x_n$.
Theorem 45 (Weyl character formula)
  1. Fix $(\pi_\lambda,V_\lambda)$ and suppose $\gamma\in T(\mathbb{R})$ is regular semisimple (i.e., $\alpha(\gamma)\ne1$ for any $\alpha\in\Phi$). Then $$\tr\pi_\lambda(\gamma)=\frac{\sum_{w\in W}\varepsilon(w)\lambda^w(\gamma)\rho^w(\gamma)}{\rho(\gamma)\prod_{\alpha\in\Phi^+}(1-\alpha(\gamma)^{-1})}.$$ Here $\rho\in X^\cdot(T_\mathbb{C})\otimes \mathbb{Q}$ is the half sum of the positive roots and $\varepsilon: W\rightarrow \{\pm1\}$ is the sign character. The denominator does not vanish by the assumption that $\gamma$ is regular semisimple.
  2. Fix $(\pi_\lambda,V_\lambda)$. Then $$\tr\pi_\lambda(1)=\dim_\mathbb{C} V_\lambda=\frac{\prod_{\alpha\in\Phi^+}\langle\lambda+\rho,\alpha^\vee\rangle}{\prod_{\alpha\in\Phi^+}\langle\rho,\alpha^\vee\rangle}.$$ $P(\lambda)=\dim_\mathbb{C} V_\lambda$ is a polynomial function.
Example 53 $G=SU_2$. Let $T=\{\gamma=\diag\{e^{i\theta},e^{-i\theta}\}\}\subseteq G$ be the diagonal torus. Then $X^\cdot(T_\mathbb{C})\cong \mathbb{Z}$, given by $\chi_n(\gamma)=e^{in\theta}$, $S=\Phi^+=\{\chi_2\}$. The dominant weights are the non-negative integers. Then $$\tr\pi_n(\gamma)=e^{in\theta}+e^{i(n-2)\theta}+\cdots +e^{-in\theta}=\frac{e^{i(n+1)\theta}-e^{-i(n+1)\theta}}{e^{i\theta}-e^{-i\theta}}.$$ This makes sense when $\chi_2(\gamma)\ne1$ (i.e, $\gamma$ is regular semisimple) and recovers exactly the Weyl character formula for $G$. Moreover, $V_n$ is the $n $-th symmetric power of the standard representation and $\dim_\mathbb{C} V_n=n+1$ is a polynomial function.
Proposition 22 Fix $\gamma\in T(\mathbb{R})$ and let $\lambda\in X^\cdot(T_\mathbb{C})$ be an $S$-dominant weight which is allowed to vary. Then $$\tr\pi_\lambda(\gamma)=\sum_{i=1}^{N_\gamma} E_i(\gamma,\lambda)P_i(\lambda).$$ Here $E_i(\gamma,\lambda)$ is a rational function of degree depending $\lambda$ with the properties:
  1. it is uniformly bounded as $\lambda$ varies;
  2. the denominator is non-zero and independent of $\lambda$.

$P_i(\lambda)$ is a polynomial function such that $\deg P_i<\deg P$, except in the case $\gamma=1$, where $P(\lambda)=\dim_\mathbb{C}V_\lambda$.

Remark 107 There are two extremal cases: when $\gamma$ is regular semisimple and $\gamma$ is trivial. The proposition simply follows from the Weyl character formula. In the first case, we let $N_\lambda=1$, $P_i(\lambda)=1$ and $E_i(\gamma,\lambda)=\tr\pi_\lambda(\gamma)$. The Weyl denominator is non-zero and does not depend on $\lambda$. The numerator is a sum of roots of unity (since $T(\mathbb{R})$ is compact), hence is uniformly bounded. In the second case, $N_\gamma=1$, $E_i(\gamma,\lambda)=1$ and $P_i(\lambda)=P(\lambda)$. For general $\lambda\in T(\mathbb{R})$, there is an argument of "descent to subgroups". See the proof in Chenevier-Clozel.
Corollary 10 Let $\lambda_1,\lambda_2,\ldots$ be a sequence of $S$-dominant weights satisfying: for any $\alpha\in S$, $\langle\lambda_i,\alpha^\vee\rangle\rightarrow\infty$ as $i\rightarrow\infty$ (we say $\lambda_i\rightarrow\infty$ far from the walls). Then for all $\gamma\in G(\mathbb{R})$, $\gamma\ne1$, we have $$\lim_{i\rightarrow\infty} \tr\pi_{\lambda_i}(\gamma)/P(\lambda_i)=0.$$
Proof After conjugation, we may assume that $\gamma\in T(\mathbb{R})$. Then $$\tr\pi_{\lambda_i}(\gamma)/P(\lambda_i)=\sum_{j=1}^{N_\gamma} E_j(\gamma,\lambda_i)P_j(\lambda_i)/P(\lambda_i).$$ Since $\lambda_i\rightarrow\infty$ far from the walls, the coefficients of the monomials of $P(\lambda_i)$ go to infinity. Because $\gamma\ne1$, $\deg P_j<\deg P$, thus $P_j(\lambda_i)/P(\lambda_i)\rightarrow 0$. Since $E_j(\gamma,\lambda_i)$ is uniformly bounded as $i$ varies, the result follows.

We now return to the trace formula. Fix a sequence $\lambda_1,\lambda_2,\ldots$ of $S$-dominant weights such that $\lambda_i\rightarrow\infty$ far from the walls. We choose $\phi_i\in C_c^\infty(G(\mathbb{A}))$ by $$\phi=\phi^{p,\infty} \otimes \phi_p \otimes \phi_\infty^{\lambda_i}.$$

We observe that

  1. If $\Pi$ is an automorphic representation of $G(\mathbb{A})$ such that $\tr\Pi(\phi_i)\ne0$ for some $i$, then $\Pi$ has the desired property: $\pi(\phi^{p,\infty})$ is simply the $K^p$ equivariant projection $\Pi^{p,\infty}\rightarrow (\Pi^{p,\infty})^{K^p}$, in particular, $\tr\Pi(\phi^{p,\infty})\ne0$ if and only if $(\Pi^{p,\infty})^{K^p}\ne0$. Since $\phi_p$ is chosen to be a matrix coefficients, $\tr\Pi_p(\phi_p)\ne0$ if and only if $\Pi_p\cong\pi_0 $. So to prove the theorem, it is enough to show $$\tr R(\phi_i)=\sum_{\Pi}m(\Pi)\tr\Pi(\phi_i)\ne0,$$ where $R$ is the regular representation of $G(\mathbb{A})$.
  2. The global orbital integral $\mathcal{O}_\gamma(\phi_i)$ splits up as a product of local orbital integrals, provided all the measures are chosen carefully; the constants are not important for the proof.

Now we apply the trace formula and obtain $$\tr R(\phi_i)=\sum_{\{\gamma\}}a(\gamma)\mathcal{O}_\gamma(\phi_i)=a(1)\mathcal{O}_1(\phi_i)+\sum_{\{\gamma\}-\{1\}}a(\gamma)\mathcal{O}_\gamma(\phi^{p,\infty})\mathcal{O}_\gamma(\phi_p)\mathcal{O}_\gamma(\phi^{\lambda_i}_\infty).$$ Notice $C=\supp(\phi_i)\subseteq K^p\supp(\phi_p) G(\mathbb{R})$ is a compact subset of $G(\mathbb{A})$, which does not depend on $i$. So by Remark 106, the sum over $\{\gamma\}-\{1\}$ can be replaced by a sum on a finite subset $S_C\subseteq \{\gamma\}-\{1\}$. Notice $\mathcal{O}_1(\phi_i)=\phi_i(1)$. We can always arrange that $\phi_p(1)\ne0$ by choosing suitable matrix coefficient. Then $$\tr R(\phi_i)=a(1)\phi_p(1)P(\lambda_i)+\sum_{\gamma\in S_C}a(\gamma)\mathcal{O}_\gamma(\phi^{p,\infty})\mathcal{O}_\gamma(\phi_p)\mathcal{O}_\gamma(\phi^{\lambda_i}_\infty).$$ Dividing by $P(\lambda_i)$, we obtain that $$R(\phi_i)/P(\lambda_i)=a(1)\phi_p(1)+\sum_{\gamma\in S_C}a(\gamma)\mathcal{O}_\gamma(\phi^{p,\infty})\mathcal{O}_\gamma(\phi_p)\mathcal{O}_\gamma(\phi_\infty^{\lambda_i})/P(\lambda_i).$$ The first term $a(1)\phi_p(1)$ is nonzero and is independent of $i$. So if we can show that $\mathcal{O}_\gamma(\phi_\infty^{\lambda_i})/P(\lambda_i)$ tends to zero, then $\tr R(\phi_i)/P(\lambda_i)\ne0$ for all sufficiently large $i$.

Let us choose a measure on $G_\gamma(\mathbb{R})\backslash G(\mathbb{R})$ so that for any $f\in C_c^\infty(G_\gamma(\mathbb{R})\backslash G(\mathbb{R}))$, $$\int_{G_\gamma(\mathbb{R})\backslash G(\mathbb{R})}f(x^{-1}\gamma x)dx=\int_{G(\mathbb{R})}f(x^{-1}\gamma x)dx.$$ This is allowable since $G_\gamma(\mathbb{R})$ is compact. So $$\mathcal{O}_\gamma(\phi_\infty^{\lambda_i})=\Vol(G(\mathbb{R}))\cdot\tr\pi_{\lambda_i}(\gamma)$$ as $\tr\pi_{\lambda_i}$ is conjugation invariant. We conclude the orbital side is nonzero since $\tr\pi_{\lambda_i}(\gamma)/P(\lambda_i)\rightarrow0$ by Corollary ##Cor:AFfarfromwalls , which finishes the proof of Theorem 44 in the compact case.


TopApplication of Arthur's simple trace formula

References for this section:

  • Chenevier-Clozel, JAMS
  • Clozel-Delorme, le theoreme de Paley-Wiener invariant pour les groupes reductifs, I, II

Today we we will remove the compact condition imposed on $G(\mathbb{R})$ last time. We are going to use Arthur's simple trace formula (Theorem 31). As last time, we choose the test functions of the form $$\phi=\phi_p\otimes\phi^{p,\infty}\otimes\phi_{\infty,\lambda}$$ and will ensure that for suitable choices of $\lambda$ and $\phi_{\infty,\lambda}$, $\mathcal{O}_\gamma(\phi_{\infty,\lambda})$ vanishes if $\gamma\in G(\mathbb{R})$ is not $\mathbb{R}$-elliptic. We may assume that $\phi_p(1)=1$ and $\Vol(K^p)=1$.

Fix $T\subseteq G_\mathbb{R}$ an $\mathbb{R}$-elliptic maximal torus. The existence of $T$ is equivalent to the existence of a compact inner $H $ form of $G$ (Theorem 25). The idea is to transfer functions (resp. conjugacy classes) on $G(\mathbb{R})$ to functions (resp. conjugacy classes) on $H(\mathbb{R})$.

We fix such an $H $ and an isomorphism $f: G_\mathbb{C}\cong H_\mathbb{C}$ such that $f^{-1}\circ \bar f\in \Aut(G_\mathbb{C})$ is an inner automorphism. Define $T_{H,\mathbb{C}}=f(T_\mathbb{C})$. The fact (one needs to know a bit more about real groups) is that one can choose $f$ such that $T_{H,\mathbb{C}}\subseteq H_\mathbb{C}$ is defined over $\mathbb{R}$ and the restriction $f|_{T_\mathbb{C}}: T_\mathbb{C}\cong T_{H,\mathbb{C}}$ is also defined over $\mathbb{R}$. Choose such an $f$ and let $T_H\subseteq H$ to be the real torus which becomes $T_{H,\mathbb{C}}$ after extension of scalars. So we have an isomorphism $f|_T: T\cong T_H$ over $\mathbb{R}$.

Example 54 Suppose $G=U(p,q)$, $H=U(n)$. Then the diagonal torus $T(\mathbb{R})=\diag\{e^{i\theta_1},\ldots, e^{i\theta_n}\}$ is an $\mathbb{R}$-elliptic torus. Choose $T_H\subseteq H$ also to be diagonal torus. Then one can find an isomorphism $f: G_\mathbb{C}\cong H_\mathbb{C}$ which restricts to an isomorphism $T_\mathbb{C}\cong T_{H,\mathbb{C}}$ defined over $\mathbb{R}$.

Recall (Theorem 26) that the square-integrable representations of $G(\mathbb{R})$ fall into packets $\Pi_\lambda$ indexed by $W$-orbits of regular elements $\lambda\in X^\cdot(T_\mathbb{C})$. Each packet $\Pi_\lambda$ contains exactly $\#W/W_\mathbb{R}$ representations with the infinitesimal character $\chi_\lambda: Z(\mathfrak{g})\rightarrow \mathbb{C}$. Similarly, the irreducible representations of the compact group $H(\mathbb{R})$ are in bijection with $W$-orbits of regular elements $\lambda\in X^\cdot(T_{H,\mathbb{C}})$. This differs from the highest weight parametrization by a $\rho$-shift (Example 33).

So another way to phrase the parametrization of discrete series of $G(\mathbb{R})$ is to put them in packets $\Pi_\sigma$, where $\sigma$ varies over irreducible representations of $H(\mathbb{R})$. The assignment $\sigma\mapsto\Pi_\sigma$ does not depend on the choice of the isomorphism $f$. This is a special case of the (local) Langlands functoriality (the $L $-groups of $G $ and $H $ are the same and $G$ is "more quasi-split" than $H $). Dual to this transfer $\sigma\mapsto\Pi_\sigma$ we should have a transfer of conjugacy classes. Let $\gamma\in G(\mathbb{R})$ be an $\mathbb{R}$-elliptic element. Then $\gamma$ is $G(\mathbb{R})$-conjugate to an element of the fixed torus $T(\mathbb{R})$. Without loss of generality, we may assume $\gamma\in T(\mathbb{R})$. We define $\gamma_H=f(\gamma)\in T_H(\mathbb{R})$, well-defined up to $H(\mathbb{R})$-conjugacy.

Theorem 46 Let $\lambda\in X^\cdot(T_\mathbb{C})$ be sufficiently regular (e.g., $\langle\lambda,\alpha^\vee\rangle\gg0$ for all $\alpha\in S$). Then there exists a function $\phi_{\infty,\lambda}\in C_c^\infty(G(\mathbb{R}))$ satisfying
  1. If $\pi$ is a unitary irreducible representation of $G(\mathbb{R})$, then $$\tr\pi(\phi_{\infty,\lambda})=
  1 & \pi\in\Pi_\lambda, \\
  0 & \text{otherwise}.
\end{cases}$$ (so $\phi_{\lambda,\lambda}$ is a pseudo-coefficient for the packet $\Pi_\lambda$).
  2. If $\gamma\in G(\mathbb{R})$ is not elliptic, then $\mathcal{O}_\gamma(\phi_{\infty,\lambda})=0$.
  3. If $\gamma\in G(\mathbb{R})$ is elliptic, then there is a sign $e(\gamma)$ such that $\mathcal{O}_\gamma(\phi_{\infty,\lambda})=e(\gamma)\tr\sigma_\lambda(\gamma_H^{-1})$ ($=e(\gamma)\tr\sigma_\lambda^\vee(\gamma_H)$).
Remark 108
  1. Notice this theorem is immediate when $G_\mathbb{R}=H_\mathbb{R}$ is compact (take $\phi_{\infty,\lambda}=\tr\sigma_\lambda$) and allows us to handle general groups by transferring to the compact group $H_\mathbb{R}$.
  2. Part a) implies Part b), c) using techniques from harmonic analysis.
Remark 109 We will not prove this difficult theorem. A proof can be found in Clozel-Delorme. Here is a rough idea. There is a natural topology (Fell topology) on the unitary dual $\tilde G$ of $G$ and the classification of tempered representation allows one to describe the tempered dual $\tilde G_t\subseteq \tilde G$ explicitly as a topological space (like disjoint union of affine spaces). There is a map $$C_c^\infty(G(\mathbb{R}))\rightarrow \mathrm{Fun}(\tilde G_t,\mathbb{C}),\quad \phi\mapsto(\pi\mapsto \tr\pi(\phi)).$$ The theorem of Clozel-Delorme characterizes the image of this map. In particular, since the discrete series representations are isolated in the tempered dual, one can pick out the discrete series representations of $G(\mathbb{R})$ using a function of the form $\tr\pi(\phi)$ (but it is not quite possible to write down such $\phi$ explicitly and $\phi$ is certainly not determined uniquely).

We can further choose all the $\phi_{\infty,\lambda}$'s to have support in a fixed compact subset of $G(\mathbb{R})$. Now applying Arthur's trace formula gives $$\tr R_0(\phi_\lambda)=\sum_{\gamma}a(\gamma)\mathcal{O}_\gamma(\phi_p)\mathcal{O}_\gamma(\phi^{p,\infty})\mathcal{O}_\gamma(\phi_{\infty,\lambda}).$$ Let $\lambda$ vary so that all $\phi_\lambda$ have support in a fixed compact subset $C\subseteq G(\mathbb{A}_\mathbb{Q})$. Then the orbital side becomes $$a(1)\mathcal{O}_1(\phi_{\infty,\lambda})+\sum_{\gamma\in S_c}a(\gamma)\mathcal{O}_\gamma(\phi_p)\mathcal{O}_\gamma(\phi^{p,\infty})\mathcal{O}_\gamma(\phi_{\infty,\lambda}),$$ where $S_C\subseteq\{\gamma\}-\{1\}$ is a finite set of $\mathbb{R}$-elliptic elements of $G(\mathbb{Q})$. But by the previous theorem, $$\mathcal{O}_\gamma(\phi_{\infty,\gamma})=e(\gamma)\tr\sigma_\lambda(\gamma_H^{-1}),$$ we obtain $$\frac{\tr R_0(\phi_\lambda)}{\dim_\mathbb{C} \sigma_\lambda}=a(1)e(1)+\sum_{\gamma\in S_C}a(\gamma)\mathcal{O}_\gamma(\phi_p)\mathcal{O}_\gamma(\phi^{p,\infty})e(\gamma)\frac{\tr(\sigma_\lambda(\gamma_H^{-1}))}{\dim_\mathbb{C}\sigma_\lambda}.$$ Let $\lambda\rightarrow\infty$ far from walls as last time, we obtain that $\tr R_0(\phi_\lambda)/\dim_\mathbb{C}\sigma_\lambda\rightarrow a(1)e(1)\ne0$ as an application of Weyl character formula. In particular, if $\lambda$ is sufficiently far from the wall, then there exists a cuspidal automorphic representation $\Pi$ such that $\tr\Pi(\phi_\lambda)\ne0$. Thus we have constructed the automorphic representation with all desired local properties in Theorem 44.