Journal of the European Mathematical Society

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Volume 10, Issue 2, 2008, pp. 351–390
DOI: 10.4171/JEMS/113

Published online: 2008-06-30

Representation growth of linear groups

Michael Larsen[1] and Alexander Lubotzky[2]

(1) Indiana University, Bloomington, United States
(2) Hebrew University, Jerusalem, Israel

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 $\calz_\Gamma(s) = \suml^\infty_{n=1} r_n(\Gamma)n^{-s}$. When $\Gamma$ is an arithmetic group satisfying the congruence subgroup property then $\calz_\Gamma(s)$ 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 $U$ 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.

Keywords: Representation growth, p-adic group, arithmetic group

Larsen Michael, Lubotzky Alexander: Representation growth of linear groups. J. Eur. Math. Soc. 10 (2008), 351-390. doi: 10.4171/JEMS/113