Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Uniformity and Functional Equations for Local Zeta Functions of
  $\mathfrak{K}$-Split Algebraic Groups   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.

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The categories of this paper are: chao-dyn
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The categories of this paper are: funct-an
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The categories of this paper are: atom-ph
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The categories of this paper are: patt-sol
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The categories of this paper are: quant-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: plasm-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: cs
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The categories of this paper are: ao-sci
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The categories of this paper are: physics
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The categories of this paper are: q-alg
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The categories of this paper are: alg-geom
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The categories of this paper are: hep-lat
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The categories of this paper are: nlin
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The categories of this paper are: dg-ga