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

TL;DR

This paper investigates the uniformity and functional equations of local zeta functions associated with algebraic groups over local fields, providing criteria for their symmetry properties and extending previous results in the field.

Contribution

It introduces new conditions under which local zeta functions of $rak{K}$-split groups are uniform and satisfy functional equations, generalizing prior work and exploring cases where these properties fail.

Findings

Local zeta functions are almost uniform for $rak{K}$-split groups with certain unipotent radical properties.

Functional equations hold for these zeta functions under specific invariants of the group and representation.

Counterexamples are constructed where the functional equations do not hold, highlighting the importance of the criteria.

Abstract

We study the local zeta functions of an algebraic group $\mathcal{G}$ defined over $\mathfrak{K}$ together with a faithful $\mathfrak{K}$-rational representation $\rho$ 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 prove that the local zeta functions are almost uniform for all $\mathfrak{K}$-split groups whose unipotent radical satisfies a certain lifting property. This property is automatically satisfied if $\mathcal{G}$ is reductive. We provide a further criterion in terms of invariants of $\mathcal{G}$ and $\rho$ which guarantees that the local zeta functions satisfy functional equations for almost all primes of $\mathfrak{K}$. We obtain these results by using a $\mathfrak{p}$-adic Bruhat decomposition of Iwahori and Matsumoto [IM] to express the zeta function as a weighted sum over the Weyl group $W$ associated to $\mathcal{G}$ 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. 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.