Uniformity and Functional Equations for Local Zeta Functions of Q-split Algebraic Groups

We study the local zeta functions of an algebraic group G defined over Q together with a faithful Q-rational representation ρ. This is given by an integral over a set of p-adic points of G determined by ρ. We prove that the local zeta functions are uniform for almost all primes p for all Qsplit groups whose unipotent radical satisfies a certain lifting property. This property is automatically satisfied if G is reductive. We provide a further criterion in terms of invariants of G and ρ which guarantees that the local zeta function will satisfy a functional equation for almost all primes. We obtain these results by expressing the zeta function as a weighted sum over the Weyl group W associated to G of generating functions over lattice points of a polyhedral cone. The functional equation reflects an interplay between symmetries of the Weyl group and reciprocities of the combinatorial object, the latter relating to a theorem of Stanley [Sta]. We construct families of groups with representations violating our second structural criterion whose local zeta functions do not satisfy functional equations. Our work generalizes results of Igusa [Igu] and du Sautoy and Lubotzky [dSL] and has implications for zeta functions of finitely generated torsion-free nilpotent groups.