JournalsjemsVol. 10 , No. 2DOI 10.4171/jems/113

Representation growth of linear groups

  • Alexander Lubotzky

    Hebrew University, Jerusalem, Israel
  • Michael Larsen

    Indiana University, Bloomington, United States
Representation growth of linear groups cover

Abstract

Let Γ\Gamma be a group and rn(Γ)r_n(\Gamma) the number of its nn-dimensional irreducible complex representations. We define and study the associated representation zeta function \calzΓ(s)=\sumln=1rn(Γ)ns\calz_\Gamma(s) = \suml^\infty_{n=1} r_n(\Gamma)n^{-s}. When Γ\Gamma is an arithmetic group satisfying the congruence subgroup property then \calzΓ(s)\calz_\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 UU of the associated simple group GG over the associated local field KK. Here we show a surprising dichotomy: if G(K)G(K) is compact (i.e. GG anisotropic over KK) the abscissa of convergence goes to 0 when dimG\dim G goes to infinity, but for isotropic groups it is bounded away from 00. 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.