Representation growth of linear groups

Larsen, Michael, and Lubotzky, Alexander. "Representation growth of linear groups." Journal of the European Mathematical Society 010.2 (2008): 351-390. <http://eudml.org/doc/277646>.

@article{Larsen2008,
abstract = {Let $\Gamma $ be a group and $r_n(\Gamma )$ the number of its $n$-dimensional irreducible complex representations. We define and study the associated representation zeta function $\mathcal \{Z\}_\Gamma (s)=\sum ^\infty _\{n=1\} r_n(\Gamma )n^\{-s\}$. When $\Gamma $ is an arithmetic group satisfying the congruence subgroup property then $\mathcal \{Z\}_\Gamma (s)$ has an “Euler factorization”. The “factor at infinity” is sometimes called the “Witten zeta function” counting the rational representations of an algebraic group. For these we determine precisely the abscissa of convergence. The local factor at a finite place counts the finite representations of suitable open subgroups $U$ of the associated simple group $G$ over the associated local field $K$. Here we show a surprising dichotomy: if $G(K)$ is compact (i.e. $G$ anisotropic over $K$) the abscissa of convergence goes to 0 when $\dim G$ goes to infinity, but for isotropic groups it is bounded away from $0$. As a consequence, there is an unconditional positive lower bound for the abscissa for arbitrary finitely generated linear groups. We end with some observations and conjectures regarding the global abscissa.},
author = {Larsen, Michael, Lubotzky, Alexander},
journal = {Journal of the European Mathematical Society},
keywords = {zeta functions; group representations; arithmetic groups; Zeta functions; group representations; arithmetic groups},
language = {eng},
number = {2},
pages = {351-390},
publisher = {European Mathematical Society Publishing House},
title = {Representation growth of linear groups},
url = {http://eudml.org/doc/277646},
volume = {010},
year = {2008},
}

TY - JOUR
AU - Larsen, Michael
AU - Lubotzky, Alexander
TI - Representation growth of linear groups
JO - Journal of the European Mathematical Society
PY - 2008
PB - European Mathematical Society Publishing House
VL - 010
IS - 2
SP - 351
EP - 390
AB - Let $\Gamma $ be a group and $r_n(\Gamma )$ the number of its $n$-dimensional irreducible complex representations. We define and study the associated representation zeta function $\mathcal {Z}_\Gamma (s)=\sum ^\infty _{n=1} r_n(\Gamma )n^{-s}$. When $\Gamma $ is an arithmetic group satisfying the congruence subgroup property then $\mathcal {Z}_\Gamma (s)$ has an “Euler factorization”. The “factor at infinity” is sometimes called the “Witten zeta function” counting the rational representations of an algebraic group. For these we determine precisely the abscissa of convergence. The local factor at a finite place counts the finite representations of suitable open subgroups $U$ of the associated simple group $G$ over the associated local field $K$. Here we show a surprising dichotomy: if $G(K)$ is compact (i.e. $G$ anisotropic over $K$) the abscissa of convergence goes to 0 when $\dim G$ goes to infinity, but for isotropic groups it is bounded away from $0$. As a consequence, there is an unconditional positive lower bound for the abscissa for arbitrary finitely generated linear groups. We end with some observations and conjectures regarding the global abscissa.
LA - eng
KW - zeta functions; group representations; arithmetic groups; Zeta functions; group representations; arithmetic groups
UR - http://eudml.org/doc/277646
ER -