JournalsjemsVol. 10, No. 2pp. 351–390

Representation growth of linear groups

  • Alexander Lubotzky

    Hebrew University, Jerusalem, Israel
  • Michael Larsen

    Indiana University, Bloomington, United States
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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.

Cite this article

Alexander Lubotzky, Michael Larsen, Representation growth of linear groups. J. Eur. Math. Soc. 10 (2008), no. 2, pp. 351–390

DOI 10.4171/JEMS/113