Geodesic
Representation growth of linear groups
Journal of the European Mathematical Society, Tome 10 (2008) no. 2, pp. 351-390 Cet article a éte moissonné depuis la source EMS Press

Voir la notice de l'article

Let Γ be a group and rn​(Γ) the number of its n-dimensional irreducible complex representations. We define and study the associated representation zeta function ZΓ​(s)=∑n=1∞​rn​(Γ)n−s. When Γ is an arithmetic group satisfying the congruence subgroup property then ZΓ​(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 dimG 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.
DOI : 10.4171/jems/113
Classification : 11-XX, 20-XX, 22-XX, 00-XX
Keywords: Representation growth, p-adic group, arithmetic group 
@article{JEMS_2008_10_2_a2,
     author = {Michael Larsen and Alexander Lubotzky},
     title = {Representation growth of linear groups},
     journal = {Journal of the European Mathematical Society},
     pages = {351--390},
     year = {2008},
     volume = {10},
     number = {2},
     doi = {10.4171/jems/113},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/jems/113/}
}
TY  - JOUR
AU  - Michael Larsen
AU  - Alexander Lubotzky
TI  - Representation growth of linear groups
JO  - Journal of the European Mathematical Society
PY  - 2008
SP  - 351
EP  - 390
VL  - 10
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.4171/jems/113/
DO  - 10.4171/jems/113
ID  - JEMS_2008_10_2_a2
ER  - 
%0 Journal Article
%A Michael Larsen
%A Alexander Lubotzky
%T Representation growth of linear groups
%J Journal of the European Mathematical Society
%D 2008
%P 351-390
%V 10
%N 2
%U http://geodesic.mathdoc.fr/articles/10.4171/jems/113/
%R 10.4171/jems/113
%F JEMS_2008_10_2_a2
Michael Larsen; Alexander Lubotzky. Representation growth of linear groups. Journal of the European Mathematical Society, Tome 10 (2008) no. 2, pp. 351-390. doi: 10.4171/jems/113

Cité par Sources :