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Uniformity and functional equations for local zeta functions of $\mathfrak {K}$-split algebraic groups
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 133, Number 1, February 2011
- pp. 1-27
- 10.1353/ajm.2011.0008
- Article
- Additional Information
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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 ${\Bbb Q}$. These are given by integrals over $\mathfrak {p}$-adic
points of ${\cal G}$ 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 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 and du Sautoy and Lubotzky and has
implications for zeta functions of finitely generated torsion-free
nilpotent groups.