Arithmetic groups have rational representation growth

Abstract

Let Γ be an arithmetic lattice in a semisimple algebraic group over a number field. We show that if Γ has the congruence subgroup property, then the number of n-dimensional irreducible representations of Γ grows like n α , where α is a rational number.

Similarity classes of integral p-adic matrices and representation zeta functions of groups of type A2

“…In the same way, it refines the variant of this conjecture that was proved in [6,Theorem 1.2]. Indeed, Corollary B asserts that, for the relevant arithmetic groups Γ, not only the degree of representation growth but also the order of the pole of the meromorphically continued function at s = α(Γ), and thus the exponent of the log-N -term in (2), are invariants of the type A 2 . Likewise meromorphic continuation can be achieved uniformly in a strip of width at least 1/6.…”

“…Otherwise the abscissa of convergence α(G) satisfies α(G) = lim sup N →∞ log (R N (G)) log N and thus gives the degree of polynomial growth. For a range of results on representation growth and zeta functions of arithmetic and profinite groups see, for instance, [41,2,30,38,51,1]. The current paper forms part of a series of papers on representation growth; see [3,4,5,6].…”

Uniform analytic properties of representation zeta functions of finitely generated nilpotent groups

“…Hence (II) follows.It is clear that δ(G) := max{a(G)−δ 1 , d 3 } is independent of the number field K and the zeta functionζ G(O) (s) may be continued to {s ∈ C | Re(s) > a(G) − δ(G)}. The order β(G) of the pole of p ∈Q V −1 p (s) at s = a(G)is clearly an invariant of G, which proves(2). The statement about the holomorphy of the continued function follows from the fact that the meromorphic continuation was achieved by writing ζG(O) (s) as a product of translates of Artin L-functions (implicit in the proof of Proposition 4.3) and a Dirichlet series convergent on {s ∈ C | Re(s) > a(G) − δ(G)}; cf.…”

“…Most general results about the analytic properties of Euler products of the nonarchimedean factors are based on suitable approximations of the individual Euler factors. Using such techniques, it was shown that the abscissa of convergence of ζ Γ (s) is a rational number ( [2]) which only depends on the absolute root system of the algebraic group G; cf. [4].…”

“…(1) The abscissa of convergence α(G) of ζ G (s) is a rational number. (2) The zeta function ζ G (s) has a meromorphic continuation to {s ∈ C | Re(s) > α(G) − δ} for some δ ∈ Q >0 . On the line {s ∈ C | Re(s) = α(G)}, the continued zeta function has a unique pole at s = α(G), of order β(G), say.…”