Zeta functions of Jordan algebras representations

Achab, Dehbia. "Zeta functions of Jordan algebras representations." Annales de l'institut Fourier 45.5 (1995): 1283-1303. <http://eudml.org/doc/75160>.

@article{Achab1995,
abstract = {This work is about a generalization of Kœcher’s zeta function. Let $V$ be an Euclidean simple Jordan algebra of dimension $n$ and rank $m$, $E$ an Euclidean space of dimension $N$, $\phi $ a regular self-adjoint representation of $V$ in $E$, $Q$ the quadratic form associated to $\phi $, $\Omega $ the symmetric cone associated to $V$ and $G(\Omega )$ its automorphism group\begin\{\}G(\Omega )=\lbrace g\in GL(V)\vert g(\Omega )=\Omega \rbrace .\end\{\}($H_1$) Assume that $V$ and $E$ have $\{\bf Q\}$-structures $V_\{\{\bf Q\}\}$ and $E_\{\{\bf Q\}\}$ respectively and $\phi $ is defined over $\{\bf Q\}$. Let $L$ be a lattice in $E_\{\{\bf Q\}\}$. The zeta series associated to $\phi $ and $L$ is defined by\begin\{\}\zeta \_L(s)=\sum \_\{l\in \Gamma \_\circ \backslash L^\{\prime \}\}[\{\rm det\}(Q(l))]^\{-s\},\forall s\in \{\bf C\}\end\{\}where $L^\{\prime \}=\lbrace l\in L\vert \{\rm det\}(Q(l))\ne 0\rbrace $, $\Gamma _\circ $ is some arithmetic subgroup of $GL(E)$. ($H_2$) Assume that $V_\{\{\bf Q\}\}$ is split, which means that its rank equals its primitive rank. The fundamental results are: 1. Under the assumptions ($H_1$) and ($H_2$) and using reduction theory (Siegel sets), we show that the zeta series $\zeta _L(s)$ converges absolutely for $\{\rm Re\}\,(s)&gt;\{N\over 2m\}$. 2. $\zeta _L$ admits an analytic continuation as a meromorphic function on the whole plane $\{\bf C\}$ and satisfies to a functional equation similar to that of Riemann’s zeta function.},
author = {Achab, Dehbia},
journal = {Annales de l'institut Fourier},
keywords = {Jordan algebra; symmetric cone; reductive group; arithmetic group; zeta function},
language = {eng},
number = {5},
pages = {1283-1303},
publisher = {Association des Annales de l'Institut Fourier},
title = {Zeta functions of Jordan algebras representations},
url = {http://eudml.org/doc/75160},
volume = {45},
year = {1995},
}

TY - JOUR
AU - Achab, Dehbia
TI - Zeta functions of Jordan algebras representations
JO - Annales de l'institut Fourier
PY - 1995
PB - Association des Annales de l'Institut Fourier
VL - 45
IS - 5
SP - 1283
EP - 1303
AB - This work is about a generalization of Kœcher’s zeta function. Let $V$ be an Euclidean simple Jordan algebra of dimension $n$ and rank $m$, $E$ an Euclidean space of dimension $N$, $\phi $ a regular self-adjoint representation of $V$ in $E$, $Q$ the quadratic form associated to $\phi $, $\Omega $ the symmetric cone associated to $V$ and $G(\Omega )$ its automorphism group\begin{}G(\Omega )=\lbrace g\in GL(V)\vert g(\Omega )=\Omega \rbrace .\end{}($H_1$) Assume that $V$ and $E$ have ${\bf Q}$-structures $V_{{\bf Q}}$ and $E_{{\bf Q}}$ respectively and $\phi $ is defined over ${\bf Q}$. Let $L$ be a lattice in $E_{{\bf Q}}$. The zeta series associated to $\phi $ and $L$ is defined by\begin{}\zeta _L(s)=\sum _{l\in \Gamma _\circ \backslash L^{\prime }}[{\rm det}(Q(l))]^{-s},\forall s\in {\bf C}\end{}where $L^{\prime }=\lbrace l\in L\vert {\rm det}(Q(l))\ne 0\rbrace $, $\Gamma _\circ $ is some arithmetic subgroup of $GL(E)$. ($H_2$) Assume that $V_{{\bf Q}}$ is split, which means that its rank equals its primitive rank. The fundamental results are: 1. Under the assumptions ($H_1$) and ($H_2$) and using reduction theory (Siegel sets), we show that the zeta series $\zeta _L(s)$ converges absolutely for ${\rm Re}\,(s)&gt;{N\over 2m}$. 2. $\zeta _L$ admits an analytic continuation as a meromorphic function on the whole plane ${\bf C}$ and satisfies to a functional equation similar to that of Riemann’s zeta function.
LA - eng
KW - Jordan algebra; symmetric cone; reductive group; arithmetic group; zeta function
UR - http://eudml.org/doc/75160
ER -