Subsection 4.1 Change of perspective: Representation theory
Definition 4.1
A unitary representation of a topological group \(G\) is a pair \((\pi, V)\) with \(V\) a Hilbert space and
\begin{equation*}
\pi\colon G\times V \to V
\end{equation*}
\begin{equation*}
g,v \mapsto \pi(g) v
\end{equation*}
continuous, where \(\pi\) acts unitarily.
Fact 4.2 A useful fact
If \(H \subseteq G\) is a closed subgroup and both are unimodular then there exists a \(G\) invariant measure on \(H\backslash G\) and so \(G \acts L^2(H\backslash G)\text{.}\)
Example 4.3
If \(G(K) = H\) and \(G(\adeles) = G\) then we can take
\begin{equation*}
L^2( G(K) \backslash G(\adeles))
\end{equation*}
\begin{equation*}
g f(x) = f(xg)\text{.}
\end{equation*}
Now if \(\pi, V\) is finite dimensional with
\begin{equation*}
V = V_1 \oplus \cdots \oplus V_n
\end{equation*}
\begin{equation*}
\pi, \chi_1,\ldots, \pi, \chi_n
\end{equation*}
then
\begin{equation*}
\RR \acts L^2(\RR) = \oint_{i\RR} e^{\pi t} \diff t\text{.}
\end{equation*}
Subsubsection 4.1.1 Automorphic forms/representations
We are concerned with
\begin{equation*}
L^2(G(\QQ) \backslash G(\adeles))
\end{equation*}
for \(G\) a reductive algebraic group \(/\QQ\text{.}\)
Heads up
\begin{equation*}
G = \GL(1)
\end{equation*}
\begin{equation*}
L^2(\QQ\units\backslash \adeles\units) = \oint
\end{equation*}
instead we would like to look at a piece of
\begin{equation*}
L^2( G(\QQ) \backslash G(\adeles))
\end{equation*}
where the center
\begin{equation*}
Z(\adeles) \hookrightarrow G(\adeles)
\end{equation*}
acts by a fixed character \(\omega\text{.}\) For \(G= \GL(1)\) this is called the Nebentypus.
\begin{equation*}
(\rho_\omega, L^2(\underbrace{G(\QQ)\backslash G(\adeles)}_{\lb G(\adeles)\rb}, \omega))\text{.}
\end{equation*}
Definition 4.4 Constant term (along a parabolic)
\begin{equation*}
f \in L^2([G(\adeles)], \omega)
\end{equation*}
then
\begin{equation*}
f_N(g) = \int_{N(\QQ)\backslash N(\adeles)} f(ng) \diff n
\end{equation*}
for example
\begin{equation*}
N_B = \begin{pmatrix} 1 \amp \cdots \amp * \\ 0 \amp \ddots \amp \vdots \\ 0 \amp 0 \amp1 \end{pmatrix}
\end{equation*}
for \(G = \GL(2)\)
\begin{equation*}
N = \begin{pmatrix} 1 \amp n \\ 0 \amp 1 \end{pmatrix}\text{.}
\end{equation*}
Definition 4.5 Cusp form
If \(f_N(g) \equiv 0\) \(\forall' g \in G(\adeles)\text{.}\)
\(\rho_\omega\) is the right regular representation on \(L^2(\lb G \rb, \omega)\text{.}\) Let \(L^2_0\) denote the space of cusp forms.
Properties:
Proposition 4.6
\begin{equation*}
L_0^2 \le L^2 \text{ is closed}\text{.}
\end{equation*}
Theorem 4.7 Gelfand-Graev-Piatetski-Shapiro
\begin{equation*}
\bigoplus_{\pi} m_{\pi} \pi = \rho_{\omega,0} = \rho_\omega|_{L^2_0}
\end{equation*}
with all \(m_{\pi} \lt \infty\) countable.
Theorem 4.8 Jacquet-Langlands
For all \(\pi\) we have \(m_\pi = 1\) for \(G = \GL(2)\) i.e.
\begin{equation*}
L^2_0 ( [\GL_2(\adeles)], \omega) = \bigoplus \pi\text{.}
\end{equation*}
Subsubsection 4.1.2 Modular forms correspond to automorphic forms
We will make a bunch of groups by letting
\begin{equation*}
\Gamma(N) = \ker( \SL_2(\ZZ) \to \SL_2(\ZZ/N\ZZ))
\end{equation*}
then we say \(\Gamma\) is a congruence subgroup if there exists \(N\) such that \(\Gamma \supseteq \Gamma(N)\text{.}\) For example
\begin{equation*}
\Gamma_0(N) = \left\{\begin{pmatrix} a \amp b \\ c \amp d \end{pmatrix} \in \SL_2(\ZZ) : c \equiv 0 \pmod N\right\}
\end{equation*}
\begin{equation*}
\Gamma_1(N) = \left\{\begin{pmatrix} a \amp b \\ c \amp d \end{pmatrix} \in \SL_2(\ZZ) : c \equiv 0 \pmod N,\,a \equiv d\equiv 1 \pmod N\right\} \lhd \Gamma_0(N)
\end{equation*}
we have an action
\begin{equation*}
\Gamma_1(N) \backslash \Gamma_0(N) \acts L^2(\Gamma_1(N) \backslash \SL_2( \RR)) = L^2(N)\text{.}
\end{equation*}
Fact 4.10 Strong approximation for \(\GL(2)\)
\begin{equation*}
G(\adeles) = G(\QQ) (G_\infty^+ \times K_O(N))
\end{equation*}
for all \(N,O\text{.}\) Where
\begin{equation*}
K_O(N) = \{g \in K : g_p \text{ upper triangular for } p|N\}
\end{equation*}
\begin{equation*}
G_\infty^+ = \{g \in \GL_2(\RR) : \det(g) \gt 0\}
\end{equation*}
\begin{equation*}
Z_\infty^+\SL_2(\RR)\text{.}
\end{equation*}
Now embed
\begin{equation*}
L^2_\chi(N) \hookrightarrow L^2(N)
\end{equation*}
via
\begin{equation*}
f \leadsto \phi_f
\end{equation*}
with
\begin{equation*}
\phi_f(\gamma(z_\infty g_\infty \times k))= \chi(k) f(g_\infty)\text{.}
\end{equation*}
From modular forms to automorphic forms
\begin{equation*}
f\in S_k(N,\chi) \leadsto \phi_f
\end{equation*}
\begin{equation*}
\phi_f\gamma(z_\infty g_\infty \times k)) = \chi(k) j(g_\infty, i)^{-k} f(g_\infty(i))\text{.}
\end{equation*}
Fact 4.11
\begin{equation*}
S_k(N,\chi) \hookrightarrow L^2_{O,\chi}(N)\text{.}
\end{equation*}
-
\begin{equation*}
K_\infty = \begin{pmatrix} \cos \theta \amp \sin \theta \\ - \sin \theta \amp \cos \theta \end{pmatrix} = \Stab(i)
\end{equation*}
\begin{equation*}
j(\gamma r(\theta), i ) = e^{-i \theta} j(\gamma, i)\text{.}
\end{equation*}
Subsubsection 4.1.3 Brief overview of local representation theory
Why?
\begin{equation*}
L^2(G(\QQ)\backslash G(\adeles)) = \bigoplus \pi \oplus \oint
\end{equation*}
\begin{equation*}
\pi = \otimes' \pi_v
\end{equation*}
for \(\pi_v\) local representations.
We will consider non-archimidean admissible representations, i.e. \((\pi, V)\) such that \(\pi^{K'}\) is finite dimensional for all \(K'\) compact.
Representations of \(\GL_2(\QQ_p)\)
Assume smooth, note that smooth implies admissible, possibly true for all \(G\text{.}\)
- 1-dimensional representations \(\chi (\det)\text{.}\)
- Principal series, let \(\chi_i = |\cdot|^{s_i} \omega, s_i \in \CC\) \(\omega \colon \QQ_p\units \to \CC\text{.}\) Let
\begin{equation*}
\chi\left( \begin{array}{cc} t_1 \amp b \\ 0\amp t_1\end{array}\right) = \chi_1(t_1) \chi_2(t_2)
\end{equation*}