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\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section]
\newtheorem*{criterion}{Criterion}

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\newtheorem{definition}{Definition}[section]


\theoremstyle{remark}
\newtheorem*{rmd}{Remark on Definition 1.2}
\newtheorem*{rmc}{Remarks on the Criterion}
\newtheorem*{rmp}{Remark on the proof of Theorem 2.2}

\begin{document}
\author{Roland Martin}
%\date{September 20, 1995}
\date{September 23, 1995}
\title {On simple Igusa local zeta functions}
\address{ Department of Mathematics, United States Naval Academy,
 Annapolis, MD 21114}
\email{rem@@sma.usna.navy.mil}

\subjclass{Primary 11D79}
\keywords{Local zeta functions, canonical bases}
%\maketitle

\def\currentvolume{1}
\def\currentissue{3}

\communicated_by{David Kazhdan} 

\setcounter{page}{108}

\begin{abstract}
The objective of this announcement is the statement of some recent results on
the classification of generalized Igusa local zeta functions associated to
irreducible matrix groups. The definition of a simple Igusa local zeta function will motivate a complete classification of certain generalized Igusa local zeta
functions associated to simply connected simple Chevalley groups. In addition to the novelty of these results are the various methods used in their proof. These methods include use of the concept of canonical basis from quantum group theory and a formula expressing Serre's canonical measure $\mu_{c}$ in terms of a suitably normalized Haar measure $\mu$ and density function $\Phi$. The relevance of these results in the general theory of Igusa local zeta functions is also discussed.
\end{abstract}

\maketitle

\section{Introduction}

The study of local zeta functions of Igusa type commenced in earnest in 1974
in \cite{I1}. Through the work of various people was developed the concept of a
generalized Igusa local zeta function. For ease of explication, only the 
specialization of this concept of interest here is introduced. 
The reader is referred to \cite{M1} or \cite{I2} for a more general definition.

Let $K$ be a finite algebraic extension of ${\bf Q}_{p}$. Let $O_{K},\;\pi O_{K}$
denote the ring of integers of $K$, ideal of nonunits of $O_{K}$, respectively.
Set $\mbox{card}(O_{K}/\pi O_{K})=q$. For technical reasons assume $2\;/{\hspace*{
-.065in}|}\hspace*{.065in}q$ (see \cite{M1}). Let $|\;\,|_{K}$ be the absolute value
on $K$ normalized as $|\pi|_{K}=q^{-1}$. Let $G^{'}$ be a simply connected
simple Chevalley $K$-group and $\rho$ a finite dimensional $K$-representation of $G^{'}$. The almost direct product $G=\rho (G^{'})({\bf G_{m}1}_{dim\,\rho})$, 
where ${\bf G_{m}}$ denotes the algebraic group defined by $K^{*}$ and ${\bf 1}_{dim\,\rho}$ the $dim\,\rho \times dim\,\rho$ identity
matrix, is a $K$-subgroup of $GL_{dim\,\rho}\,$, not contained in $SL_{dim\,\rho}\,$. Set $T$ as the maximal $K$-split torus in $G$, $f$  as the positive generator of $\mbox{Hom}(G, {\bf G_{m}})$ and $G^{0}=G_{K} \bigcap \mbox{Mat}_{dim\,\rho}(O_{K})$ where $G_{K}$ denotes the $K$-rational points of $G$.

%\vs

%\noindent {\bf DEFINITION 1.1:} 
\begin{definition}
The generalized Igusa local zeta function associated to
$(G^{'},\rho)$ is
$$Z_{K}(s)=\int_{G^{0}}|f(g)|_{K}^{s}\mu_{c}(g)$$
where $\mu_{c}$ is Serre's canonical measure defined on $G_{K}$ and $s \in {\bf C},\;\mbox{Re}(s)>0$.
\end{definition}
%\vs

Typically, one would like to compute a finite form for $Z_{K}(s)$ and examine
this finite form for further relations. For example, the generalized Igusa local zeta
function associated to $(SL_{1},1)$ has finite form $Z(q^{-1},q^{-s})
=(1-q^{-1})/(1-q^{-s-1})$ with relation
$$Z(q^{-1},q^{-s})|_{q \mapsto q^{-1}}=q^{-s}Z(q^{-1},q^{-s})\,.$$
Note that this generalized Igusa local zeta function is universal in the
variables $q^{-1},\,q^{-s}$ (see \cite{I2}). 
There is a criterion for answering the typical questions in the case of a generalized Igusa local zeta function associated to $(G^{'}, \rho)$.

Since the measure $\mu_{c}$ is difficult to work with, define a function
$\Phi$ on $G_{K}$ by
$$\Phi(g)=\frac{\mu_{c}(g(G_{K} \bigcap GL_{dim\,\rho}(O_{K})))}{\mu(G_{K} \bigcap
GL_{dim\,\rho}(O_{K}))}$$
where $\mu$ is the Haar measure on $G_{K}$ normalized to be canonical measure on the group $G_{K} \bigcap GL_{dim\,\rho}(O_{K})$. It follows that $\mu_{c}(g)=
\Phi(g)\mu(g)$ and, thus, that
$$Z_{K}(s)=\int_{G^{0}}|f(g)|_{K}^{s}\Phi(g)\mu(g)\,.$$
The details are in \cite{M1}. Via the $p$-adic Bruhat decomposition for $G_{K}$,
the computation of $\Phi(g)$ is reduced to that of $\Phi(\xi(\pi))$ where
$\xi \in \mbox{Hom}({\bf G_{m}},T)$ is as in \cite{I2} or \cite{M1}.

%\vs

%\noindent {\bf DEFINITION 1.2:} 
\begin{definition}
The generalized Igusa local zeta function
$Z_{K}(s)$ associated to $(G', \rho)$ is {\it simple} if $\Phi$ is
independent of $\xi$.
\end{definition}
%\vs

%\noindent {\bf REMARK} on {\bf DEFINITION 1.2}: 
\begin{rmd}
As just explained above, 
$Z_{K}(s)=Z_{K}(s,\Phi)$. Therefore, the property of $Z_{K}(s)$ being a 
simple generalized Igusa local zeta function depends only on $Z_{K}(s)$ 
itself.
\end{rmd}
%\vs    

%\noindent {\bf CRITERION:}([4]) 
\begin{criterion}[{\cite{I2}}]
If $Z_{K}(s)$ is simple, then $Z_{K}(s)$
has a finite form that expresses $Z_{K}(s)$ as a rational function $Z(q^{-1},
q^{-s})$ satisfying the functional equation %%\\[.28in]
%$\bullet$\hspace*{1.40in}$Z(q^{-1},q^{-s})|_{q \mapsto %q^{-1}}=q^{-deg(f)s}Z(q^{-1},q^{-s}).$
%$\bullet\hfill
\begin{equation} 
Z(q^{-1},q^{-s})|_{q \mapsto q^{-1}}=q^{-deg(f)s}Z(q^{-1},q^{-s}).
\tag*{$\bullet$}
\end{equation}
%\hfill$
%\vskip.13in
\end{criterion}
%\noindent {\bf REMARKS} on the {\bf CRITERION}: 
\begin{rmc}
Note first that $Z_{K}(s)$ is
universal in the variables $q^{-1}$, $q^{-s}$. Note also that $\Phi$ is 
not ``residual"
as defined by J. Denef and D. Meuser because $\Phi(g)$ does not only depend
on $g\;\mbox{mod}\,\pi O_{K}$. Therefore, it is not immediate that
$Z_{K}(s)$ has either a finite form or the functional equation $\bullet$
(see \cite{DM}).
\end{rmc}

\section{Results}

There is now motivation for classifying as simple or not the generalized Igusa 
local zeta functions associated to $(G^{'},\rho)$. By assumption, $G^{'}$ is in
one of the families of nine simply connected simple Chevalley $K$-groups distinguished by their
respective Lie algebras 
$A_{\ell},\,B_{\ell},\,C_{\ell},\,D_{\ell},\,E_6,\,E_7,\,E_8,\,F_4$ and $G_2$.
Denote by ${\bf L}G^{'}$ the Lie algebra of $G^{'}$. 
It is well known that the nodes of the Dynkin diagram of ${\bf L}G^{'}$  
may be labeled canonically by the fundamental representations of $G^{'}$. We denote these fundamental representations by $\pi_{i}$, $1 \leq i \leq rk\,{\bf L}G^{'}$. Recall that $\rho$ has a unique expression in terms
of the fundamental representations $\pi_{i}$ as

%\vs

\begin{displaymath}
\rho = {{{\scriptstyle{rk\;{\bf L}G'}}\atop *}\atop{\scriptstyle{i=1}}}\pi_i^{n_i}
\end{displaymath}

%\hspace*{2.52in}$\rho=\;\stackrel{rk\;{\bf L}G^{'}}*\pi_{i}^{n_{i}}$

%\vspace*{-.1in}{\hspace*{2.92in}$\scriptstyle{i=1}$}
%\vspace*{.1in}

\noindent where $\ast$ denotes the Cartan product. We are able to parametrize
the simple generalized Igusa local zeta functions associated to $(G^{'},\rho)$
by examining the above Cartan product decomposition of $\rho$.

Therefore, orient the Dynkin diagrams of the simple Lie algebras
as in \cite{V} and label their nodes consecutively with the $\pi_{i}$ from left 
to right reserving for a branched node the last index $rk\,{\bf L}G^{'}$. For example, the Dynkin diagram for $D_{\ell}$ would be labeled

%\vs

\noindent\hskip-.5cm\begin{picture}(300,10)(10,55)
\put(120,10){\circle*{4}}
\put(115,0){$\pi_{1}$}
\put(120,10){\vector(1,0){32}}
\put(150,10){\circle*{4}}
\put(145,0){$\pi_{2}$}
\put(150,10){\vector(1,0){32}}
\put(180,10){\circle*{4}}
\put(175,0){$\pi_{3}$}
\put(190,10){\ldots\ldots}
\put(229,10){\circle*{4}}
\put(224,0){$\pi_{\ell-3}$}
\put(229,10){\vector(1,0){32}}
\put(259,10){\circle*{4}}
\put(254,0){$\pi_{\ell-2}$}
\put(259,10){\vector(0,1){32.5}}
\put(259,40){\circle*{4}}
\put(254,50){$\pi_{\ell}$}
\put(259,10){\vector(1,0){32}}
\put(289,10){\circle*{4}}
\put(284,0){$\pi_{\ell-1}$}
%\put(315,0){.}
\end{picture}

\vskip1in

%\noindent {\bf DEFINITION 2.1:} 
\begin{definition}
Given $(G^{'},\rho)$, the subgraph of the Dynkin diagram of $G^{'}$ determined by the elimination of any nodes (and 
their incident edges) associated to fundamental representations $\pi_{i}$ of
$G^{'}$ not occurring in the Cartan product decomposition of $\rho$ is called
the {\it decomposition diagram}.
\end{definition}
%\vs  

Our result is the following.

%\vs

%\noindent {\bf THEOREM 2.1:} ([7]) 
\begin{theorem}[{\cite{M3}}]
Let $G^{'}$ be of type $A_{\ell}\,(\ell\geq 1),\,B_{\ell}\,(\ell \geq 2),\,C_{\ell}\,(\ell\geq 3),\,D_{\ell}\;(\ell \geq 4),\,E_6,\,E_{7},\,E_{8},\,F_{4}$ or $G_{2}$. The  generalized Igusa local zeta function $Z_{K}(s)$ associated to $(G^{'},\rho)$ is simple if and only if
the decomposition diagram of $(G^{'},\rho)$ is connected.
\end{theorem}
Thus, in the case of $G^{'}$ of type $D_{\ell}$ and 
$\rho={{{\ell}\atop{*}}\atop{\scriptstyle{i=1}}}\pi_{i}^{n_{i}}\,$, 
the corresponding $Z_{K}(s)$ is simple
%\vspace*{-.1in}{\hspace*{2.81in}$\scriptstyle{i=1}$}
unless one or more of the following conditions hold:

%\vs

\indent (1) there exist $n_{i'},\,n_{j'}$ nonzero, $1 \leq i'1$ and\\
\indent $n_{i'+1}=\ldots=n_{j'-1}=0$, or

%\vs 

\indent (2) there exists $n_{i'}$ nonzero,
$1 \leq i' < \ell-2$, such that $n_{\ell-1},\,n_{\ell}$ 
or  $n_{\ell-1}\;\mbox{\rm and}\;n_{\ell}$\\
\indent are nonzero and $n_{i'+1}=\ldots=n_{\ell-2}=0$.

%\vs

The main intermediary result, % {\bf THEOREM 2.2} 
Theorem 2.2 below, is proved using the concept of canonical basis from 
quantum group theory. The reader is referred
to \cite{K} for background.

%\vs

%\noindent {\bf THEOREM 2.2:} ([7]) 
\begin{theorem}[{\cite{M3}}]
Let $\rho$ be a finite dimensional irreducible $K$-representation of $G^{'}$. 
Suppose the $Z_{K}(s)$ corresponding
to $(G^{'},\rho)$ is simple. Then the $Z_{K}(s)$ corresponding 
to $\left(G^{'},\;{{{\scriptstyle{n}}\atop{*}}\atop{\scriptstyle{i=1}}}\rho
\right)$ is simple.
\end{theorem}
%\vspace*{-.1in}{\hspace*{.96in}$\scriptstyle{i=1}$}

%\vs

%\noindent {\bf REMARK} on the proof of {\bf THEOREM 2.2:} 
\begin{rmp}
The proof is an induction argument taking advantage of various properties of 
the canonical basis
of the representation space $V$ of $\rho$.
\end{rmp}
%\vs

Use of %{\bf THEOREM 2.2} 
Theorem 2.2 allows one to prove %{\bf THEOREM 2.1} 
Theorem 2.1 via case-by-case examination. 

One can see that most choices of $(G^{'},\rho)$ yield non-simple $Z_{K}(s)$.
There are some isolated results related to determining the finite form and 
verifying the functional equation $\bullet$ for $Z_{K}(s)$ in the non-simple 
case (see \cite{M2}). It is, however, unknown which of these $Z_{K}(s)$ have 
finite forms and satisfy $\bullet$ .

%\input{AnnounceReferences.tex} 

%\end{document}
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%Begin-File: AnnounceReferences.tex
\bibliographystyle{amsalpha}
\begin{thebibliography}{8}

\bibitem{DM} J. Denef and D. Meuser, {\it A functional equation of Igusa's
local zeta function}, AJM {\bf 113} (1991), pp. 1135-1152. 
%\MR{93e:11145}

\bibitem{K} M. Kashiwara, {\it Crystalizing the $q$-analogue of universal
enveloping algebras}, Comm. Math. Phys. {\bf 133} (1990), pp. 249-260.
%\MR{93b:17018}

\bibitem{I1} J. Igusa, {\it Complex powers and asymtotic expansions} I,
J. reine. angew. Math. {\bf 268/269} (1974), pp. 110-130; ibid {\bf 278/279}
(1975), pp. 307-321.
%\MR{50:254}; \MR{53:8018}

\bibitem{I2} J. Igusa, {\it Universal $p$-adic zeta functions and their
functional equations}, AJM {\bf 111} (1989), pp. 671-716.
%\MR{91e:11142}

\bibitem{M1} R. Martin, {\it On generalized Igusa local zeta functions 
associated to simple Chevalley $K$-groups of type $A_{\ell},\;B_{\ell},\;
C_{\ell},\;D_{\ell},\;E_{6},\;E_{7},\;E_{8},\;F_{4}$ and $G_{2}$ 
under the adjoint representation}, preprint 1992.

\bibitem{M2} ---------, {\it The universal $p$-adic zeta function associated 
to the adjoint group of $SL_{\ell +1}$ enlarged by the group of scalar 
multiples}, preprint 1992.

\bibitem{M3} ---------, {\it On the classification of Igusa local zeta 
functions associated to certain irreducible matrix groups}, preprint 1995.

\bibitem{V} V. Varadarajan, {\it Lie groups, Lie algebras, and their
representations}, Springer-Verlag, New York, 1984.
%\MR{85e:22001}

\end{thebibliography}
\end{document}