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We study the local zeta functions of an algebraic group $\mathcal{G}$ definedover $\mathfrak{K}$ together with a faithful $\mathfrak{K}$-rationalrepresentation $\rho$ for a finite extension $\mathfrak{K}$ of $\mathbb{Q}$.These are given by integrals over $\mathfrak{p}$-adic points of $\mathcal{G}$determined by $\rho$ for a prime $\mathfrak{p}$ of $\mathfrak{K}$. We provethat the local zeta functions are almost uniform for all $\mathfrak{K}$-splitgroups whose unipotent radical satisfies a certain lifting property. Thisproperty is automatically satisfied if $\mathcal{G}$ is reductive. We provide afurther criterion in terms of invariants of $\mathcal{G}$ and $\rho$ whichguarantees that the local zeta functions satisfy functional equations foralmost all primes of $\mathfrak{K}$. We obtain these results by using a$\mathfrak{p}$-adic Bruhat decomposition of Iwahori and Matsumoto [IM] toexpress the zeta function as a weighted sum over the Weyl group $W$ associatedto $\mathcal{G}$ of generating functions over lattice points of a polyhedralcone. The functional equation reflects an interplay between symmetries of theWeyl group and reciprocities of the combinatorial object. We construct familiesof groups with representations violating our second structural criterion whoselocal zeta functions do not satisfy functional equations. Our work generalizesresults of Igusa [Igu] and du Sautoy and Lubotzky [dSL] and has implicationsfor zeta functions of finitely generated torsion-free nilpotent groups.

Uniformity and Functional Equations for Local Zeta Functions of $\mathfrak{K}$-Split Algebraic Groups