## Tuesday, August 22, 2006

For the above Axiomatization of the Example in Garrett's main vignette on the subject (the rank one case derivation, from Bernstein's general theory), you will have to add the declaration
\usepackage{color}
to your preamble. Otherwise it should compile in AMS enabled Latex.

\textbf{Axiomatization of the Example" of $\mathrm{SL}(2)$
given at the conclusion of the vignette".}
Preliminaries: notation.
$L^2(\Gamma\backslash G/K,\ell):=\left\{f\;\text{is locally integrable on \Gamma\backslash G/K so that}\; \int_{\Gamma\backslash G} |f(x)|^2a_x^{-2\ell}\intd x<+\infty\right\}.$
We will also call $L^2(\Gamma\backslash G/K)$ $V"$ or $V_{\ell}$, because
it is the domain Banach space for the system $X_s$.
\vspace*{0.2cm}

\noindent\textbf{Define}
$E_s(g)=E(a_g^s).$
The series expression for $E_s$ converges absolutely and uniformly
on compacta in $g\in G$ and $s\in\mathbf{C}$ for $\mathrm{Re}(s) \geq \sigma_0>1$.\vspace*{0.2cm}

\noindent {\color{red} First input to theory:} $E_s$
is a function of $\textbf{moderate growth}$ meaning that for
large $N$,
$a^{-N}E_s\;\text{is bounded on every Siegel set S_t.}$\vspace*{0.2cm}

\noindent {\color{red} Second input:} Computation (rough!) of the constant
term--consisting of the

\noindent \textbf{Proposition} The constant term $c_PE_s$ of $E_s$
is of the form
$c_PE_s=a^s+ba^{1-s}$
for a constant $b$ depending on $s$. (We would like to
redo this proof without using adeles.)\vspace*{0.2cm}

\noindent {\color{red} Third input:} By the \textit{Theory
of the Constant Term} (as exposed in another Garrett vignette")
$E_s\in L^2(\Gamma\backslash G/K,\ell)$ if $\ell$ is large enough. Specifically,
what is needed is proved in the Proposition and Corollary on
P. 6 of the Constant Term" vignette.\vspace*{0.2cm}

\noindent{\color{red} Fourth input} The eigenvalue relation
$\eta*E_s=\lambda_s E_s,$
where
\begin{itemize}
\item $\eta$ is (any) $K$-conjugation invariant test function on $G$.
\item $\lambda_s$ is a function of $s$, depending on $\eta$, of course.
\item $\lambda_s$ can be shown to be an \textit{entire} function of
$s\in\mathbf{C}$, and $\eta$ can be chosen so that $\lambda_s$
is \textit{non-constant}, and in particular not identically $0$,
and given so there is a choice of $\eta$ so that $\lambda_{s_0}\neq 0$.
\end{itemize}\vspace*{0.2cm}

\noindent{\color{red}Important remarks.} In our version we will
want to avoid all this abstraction by doing computations with a specific
(Gaussian) $\eta$. In EGM's version they appear to use the Selberg
transform" introduced in Chapter 3 to make the relation between
$\eta$ and $\lambda_s$ more explicit. So in any case
this part of the argument can be bypassed.\vspace*{0.2cm}

\noindent\textbf{Definition.} $X_s$ is the holomorphically
parametrized system of equations in $L^2(\Gamma\backslash G/K,\ell)$
\begin{itemize}
\item[(1)] $\left(a\frac{\partial}{\partial a}-(1-s)\right)\cdot c_Pv_s=(2s-1)a^s$, as distributions.
\item[(2)] $(\eta-\lambda_s)v_s=0$, for all $\eta\in C_c^{\infty}(G)^ {\rm inv}$.
\end{itemize}\vspace*{0.2cm}

\noindent{\color{red} Important remarks.} Presumably (2)
will be replaced by the condition that Eisenstein series
are eigenfunctions of convolution with a specific family of Gaussians;
(1) must be determined in an \textit{ad hoc} fashion following
the rough" computation of the constant term.\vspace*{0.2cm}

\noindent{\color{red} Combination of Second, Third, Fourth
inputs gives:} For $\mathrm{Re}(s)\gg 0$, $X_s$ has at least
the solution $E_s$.\vspace*{0.2cm}

\noindent{\color{red} This range of convergence, \textit{i.e.} the region
$\mathrm{Re}s\gg 0$, is precisely where (as will be shown)
$E_s$ is the unique solution for $X_s$.}\vspace*{0.2cm}

\noindent{\color{red} Fifth input.} The system $X_s$ has the
structural property that if $f$ is a difference
of solutions to $X_s$ then $f$ is a left $(\Gamma\cap N)$-invariant
function of moderate growth on $(\Gamma\cap N)\backslash G$
with $\eta f=f$ for some $\eta\in C_c^{\infty}(G)^{\rm inv}$.\vspace*{0.2cm}

\noindent {\color{red}The fifth input and the theory of the constant term}
imply that if $v_1$ and $v_2$ are solutions to $X_s$ in the range of
convergence, then $v_1-v_2\in L^2(\Gamma\backslash G/K)\cap L^2(\Gamma\backslash G)$. {\color{red} The fourth input} implies
that $v_1-v_2$ is an eigenvalue of convolution with the heat
gaussian with nonconstant
eigenfunction, holomorphic in $s$ (which dependence can
be determined explicitly). Since $v_1-v_2\in L^2(\Gamma\backslash G/K)$,
we can apply a Hilbert space argument to show that $v_1-v_2\equiv 0$.
{\color{blue} This completes the proof of uniqueness
(in the range of convergence $\mathrm{Re}(s)\gg0$)}.\vspace*{0.2cm}

\noindent
{\color{red}Sixth input} There exists an operator
on the Banach space $V$, called \texttt{trunc} such that
for any $\eta\in C_c^{\infty}(K\backslash G)$,
\begin{itemize}
\item[(a)] The operator $\eta\circ\texttt{trunc}$ on $V$ is compact.
\item[(b)] $\texttt{trunc}(\eta*h)=\eta*(\texttt{trunc}\eta)$
\end{itemize}
\vspace*{0.2cm}

\noindent
{\color{red} Seventh input} The system $X_s$ has the property that
for $v\in V_{\ell}$ a solution for the $s$ close to some fixed $s_0$,
$c_Pv=c_1a^s+c_2a^{1-s},\;\text{for some constants c_1, c_2 poss. depending on s.}$
In this context, constant" means not depending on the $a$, in other words the
$G$-variable. Further, the above inequality must hold
for $s$ \textit{away from finitely many points}.
\vspace*{0.2cm}

\textbf{Remark.} The proof of the seventh input is elementary; it
follows from the observation that $a^s$ and $a^{1-s}$ are two
linearly independent solutions of the differential equation
required of the constant term.\vspace*{0.2cm}

\noindent
{\color{red} Eighth input} The system $X_s$ and the operator
$\texttt{trunc}$ together have the property that
for $v\in V_{\ell}$ a solution for the $s$ close to some fixed $s_0$,
$\texttt{trunc}(v)\;\text{is rapidly decreasing}.$
\vspace*{0.2cm}

\noindent
{\color{red} Ninth input} The holomorphy of the function
$s\mapsto \texttt{trunc}(a^s),$
which is in fact entire by a direct computation.
\vspace*{0.2cm}

\noindent
\textbf{Definitions relating to the dominant system.}
\begin{itemize}
\item
$L^2_a(\Gamma\backslash G/K,\ell)=\{f\in V_{\ell}\;|\; \texttt{trunc}f=f\}$;
\item
$V'=\mathbf{C}\oplus\mathbf{C}\oplus L^2_a(\Gamma\backslash G/K,\ell)$;
\item
$T_s: V'\rightarrow V_{\ell}$ given by $T_s(b,c,h)=\left(\Id_{V_{\ell}}- \texttt{trunc}\right)(ba^s+ca^{1-s})+h$;
\item
$T_s': V'\rightarrow V_{\ell}$ given by $T_s'=(\eta-\lambda_s)\circ T_s$.
\item
$A: V_{\ell}\rightarrow V'$ given by $Av=(0,0,\texttt{trunc}v)$.
\end{itemize}

\noindent {\color{blue} Assembly of proof from these inputs.} We wish
to use the proposition on Dominance of linear systems of equations
found on the top of page 7 in Garrett's notes.
Everything is obvious or follows from Bernstein's general theory
(first three sections of Garrett notes)
from the definitions except that a) $X'$
is itself a \textit{holomorphically} parametrized system of equations
and b) that $X'$ is locally finite. For a) we use
the {\color{red}ninth input} to show that the family of linear maps
$\mathbf{C}^2\rightarrow L^2(\Gamma\backslash G/K,\ell),\; \text{defined by}\; (b,c)\mapsto E(\mathrm{tail}(ba^s+ca^{1-s}))$
is holomorphic in $s$. Since the restriction to $L^2_a(\Gamma\backslash G/K)$ of $T_s$ is just the inclusion map, and does not depend on $s$,
we can now conclude that $s\mapsto T_s$ is holomorphic. Then
we use the {\color{red}fourth input}, specifically the last
point about $\lambda_s$ being holomorphic, to conclude that
$T_s'$ is holomorphic.
\vspace*{0.2cm}

\noindent In order to confirm b), that $X'$ is locally finite,
we wish to apply Bernstein's {\color{blue}compact operator
criterion}, the Corollary in the upper part of page 8 of
Garrett's notes. With $A$ as above, we calculate that
for $h\in L^2_a(\Gamma\backslash G/K,\ell)$,
\begin{aligned} A\circ T'_{s_0}(h)&=&& \texttt{trunc}(\eta_{s_0}*h)-\texttt{trunc}(\lambda_{s_0}\cdot h)\&=&&\texttt{trunc}(\eta_{s_0}*h)-\lambda_{s_0} \texttt{trunc} (h)\&=&&\eta_{s_0}\circ\texttt{trunc}(h)-\lambda s_0 (h), \end{aligned}
where in the second line we have used simply that $\lambda_{s_0}$ is a
constant (\textit{i.e.}, independent of the $G$-variable),
while in the third line we have used {\color{red}part (b) of the sixth
input}. So we have shown that $(-1/\lambda_{s_0})A$ is an inverse for $T_{s_0}'$
on $L^2_a(\Gamma\backslash G/K,\ell$), modulo a scalar multiple of
the operator $\eta\circ\texttt{trunc}$, and we now apply {\color{red}part (a) of the
sixth input} and use the fact that $L^2_a(\Gamma\backslash G/K,\ell)$
is of finite codimension in $V'$ to conclude that $A$ is an inverse for $T_{s_0}'$
modulo a compact operator. This allows the use of the compact operator criterion
and finishes the proof of completeness.

NB: this exposition is still incomplete because we haven't
mentioned the source of the fact that $\lambda_{s_0}$ is a \textit{nonzero} scalar,
which is used in the last stage of the argument for the application
of the Compactness Criterion. Maybe some other things...?