## 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...?

## Friday, August 18, 2006

### Section 1 of Garrett's notes

I have put together the following brief summary of what I believe to be most important from section 1 of Garrett's notes on meromorphic continuation, "Weak-to-Strong Principles", along with some notes on the needed background and a few questions I still have about loose ends. I have TeXed it without macros, so if you cut-and-paste into your favorite editor, it should compile as an AMSLaTeX document without trouble. (If there is trouble, let me know.) This seems to be the best way of dealing with the fact that Blogger doesn't recognize any MathML.

textbf{Comments on the first section of notes, Meromorphic continuation
of Eisenstein Series for $\mathrm{SL}(2)$, April 2, 2001, Weak-to-strong
Principles"}
The main part begins with the definition of quasicomplete TVS"
(TVS"=Topological vector space).

\textbf{Quasicomplete TVS}: One for which every bounded Cauchy net
is convergent.

Reminder from your favorite topology book, \textit{e.g.}, \textit{Munkres}:
\begin{itemize}
\item
a \textbf{net in $X$} is a function from a directed set $J$ into $X$.
\item $(x_{\alpha})\rightarrow x$ if $\forall U$ neighborhood
of $x$, $\exists\alpha\in J$ such that $\alpha\leq \beta\Rightarrow x_{\beta}\in U$.
\item The definition of a Cauchy net in a TVS can then be reconstructed.
\end{itemize}

\textbf{Proposition.}\hspace*{0.3cm} Continuous compactly supported function $f: X\rightarrow V$
with values in a \textbf{quasicomplete} (locally convex) TVS $V$
have GP [=Gelfand-Pettis=vector-valued] integrals with respect to finite
positive regular Borel measures $\mu$ on compact spaces $X$,
and these integrals are unique.

\begin{proof}\textit{Overview of the Proof.} The proof has two parts.
\begin{itemize}
\item[*] $\forall K\subset V$,
$\overline{\mathrm{Co}(K)}$ is compact.
\end{itemize}
\begin{itemize}
\item[(i)] If $V$ has Property $(*)$, then the conclusions hold. This
is shown, essentially, in the proof Theorem 3.27 of Rudin, although the statement
of the Theorem is less general.
\item[(ii)] Every quasi-complete space has Property $(*)$. I take
it this follows
from the strengthened form of the Krein-Millman theorem found as
Theorem 3, $\S \mathrm{IV}.5$ of Bourbaki's \textit{TVS's}, but
I have not checked this point in detail.
\end{itemize}
\end{proof}

Remembering the following formula is useful for differentiating under
the GP integral sign. Under suitable hypotheses (stated precisely
by Garrett) which will always be valid in our applications we have
$T\left(\int_X f(x)\,\text{d}\mu(x)\right)=\int_X Tf(x)\, \text{d}\mu(x).$
Here,
\begin{itemize}
\item $T: V\rightarrow W$ (continuous linear map).
\item $f: X\rightarrow V$ (continuous compactly supported $V$-valued
function).
\item $\mu$ (finite positive Borel measure).
\end{itemize}

\textbf{Corollary} (of the first proposition) Let $V$ be a quasicomplete
(and locally convex) TVS. Then weakly holomorphic $V$-valued
functions are strongly holmorphic.
\begin{proof} The pattern for the proof is the same as that
of part (c) of Rudin's Theorem 3.31 from part (b) of the same theorem.
The only difference is that we have weakened his hypothesis that $V$
($X$") is a complete Fr\'{e}chet space; used here, is the fact that
in Garrett's result on the existence of GP integrals, we have used
only one property of $V$: quasicompleteness.
\end{proof}

\textit{The next corollary seems to belong after a proposition" which doesn't
exist in the text.} It would be nice to have a reference at least,
but Garrett provides none, and I am so far unable to find one either.

In order to understand this corollary, we need to recall what a topology
induced from seminorms is.

\textbf{Theorem 1.37} Suppose $\mathscr{P}$ is a separating
($\forall x\neq 0$, $\exists p\in \mathscr{P}$ with $p(x)\neq 0$)
family of seminorms on a vector space $X$. Associate to each
$p\in\mathscr{P}$ and to each positive integer $n$
the set
$V(p,n)=\{x:\, p(x)<1/n\}.$
Let $\mathscr{B}$ be the collection of all finite intersections of the
sets $V(p,n)$. Then $\mathscr{B}$ is a convex balanced local base
for a topology $\tau$ on $X$, which turns $X$ into a locally convex
vector space such that
\begin{itemize}
\item[(a)] every $p\in\mathscr{P}$ is continuous.
\item[(b)] a set $E\subset X$ is bounded if and only if every
$p\in\mathscr{P}$ is bounded on $E$.
\end{itemize}

\textit{Question/Exercise}: Is the topology induced on $\mathrm{Hom}^0(X,Y)$
\textit{by the family of seminorms $\rho_{x,\mu}(T)=|\mu(T(x))|$}
the same as the weak*-topology?

\textbf{Corollary.}\hspace*{0.5mm} A weakly holomorphic (means
all $\mathbf{C}$-valued function $s\mapsto\mu(T_S(x)))$ are holomorphic)
$\mathrm{Hom}^0(X,Y)$-valued function (where $X$ is an LF-space
and $Y$ is a quasicompact space) is (strongly) holomorphic.

Here, (strongly) holomorphic refers to the existence of the limit
in the weak operator topology induced by the family of seminorms
$\rho_{x,\mu}(T)=|\mu(T(x))|.$

## Wednesday, August 16, 2006

### Background for Garrett

So far all I have determined for sure concerning the Garrett exposition is that one needs more background than I have on topological vector spaces and vector-valued integration/holmorphic functions. It looks so far that the most relevant references are as follows:

Bourbaki: Integration, Chapter VI, Integration Vectorielle

________: Topological Vector Spaces, Chapter III, Spaces of Continuous Linear Maps, esp. III.8, "...Quasi-complete spaces"

Rudin: Functional Analysis, Chapter 3, "Convexity", esp. the last sections on Vector-Valued Integration and Weak-Versus Strong Holmorphicity of vector-valued functions on open domains of C.

I've read most of the last and found it to be of considerable help in interpreting Garrett's cryptic comments, but haven't looked much at the Bourbaki yet.

### Garrett's exposition of Bernstein's method

I am going to look at Garrett's exposition of Bernstein's method, since it seems easier to understand and has a few more references to the literature for background info. It can be found at the following location:

http://www.math.umn.edu/~garrett/m/v/

Then go to

[ Meromorphic continuation of Eisenstein series ][pdf] ... [ updated 02 Apr '01] ... after Bernstein-Selberg. Revised setup and treatment of SL(2,Z). Draft.

## Friday, August 11, 2006

### Treatment of the Cuspidal Elliptic (Regular Cuspidal) contribution to the trace

Elstrodt Mennike and Grunewald calculate the contribution to the trace from the Cuspidal Elliptic elements in Chapter 6, section 5 of their book. What I don't understand is the transition from equation (5.19) to (5.20) in their treatment.

Trying to match up the steps in their calculation with those of the Jo-La manuscript is a bit difficult, because Jo-La use a lot of facts that are particular to the special case of $\Gamma=SL(2,Z)$ in this section. However, it seems that this step roughly corresponds to the proof on the top of p. 385 in the manuscript. (This is in part 5, chapter 13, towards the end of section 3.) The trouble is that the details in that proof are not filled in, and we are referred back to the details of proofs in section 1. More on this later...

### Bernstein's Lectures on Eisenstein Series: Questions on Lecture 1

They are linked from the following page:

http://www.math.uchicago.edu/~arinkin/langlands/

Questions on Lecture 1
1. What is meant by the last sentence in the remark on p. 2? In particular, what does the notation $f_{\lambda}=\exp(\lambda,j)$, where $j$ is $j$-invariant" mean?

2. What does Section 4. reduce to in the case of SL(2,R)?

3. What is the modified action of $L_p$ on $F(y_p)$ in the SL(2,R) case?

4. What does $Z(L_p)\supset Z(G)$ look like concretely in the case of $G=SL(n,R) and$P\$ a standard parabolic?

5. Why does reduction theory imply the Proposition on p. 7?