Uniform cell decomposition with applications to Chevalley groups

Abstract

We express integrals of definable functions over definable sets uniformly for non-Archimedean local fields, extending results of Pas. We apply this to Chevalley groups, in particular proving that zeta functions counting conjugacy classes in congruence quotients of such groups depend only on the size of the residue field, for sufficiently large residue characteristic. In particular, the number of conjugacy classes in a congruence quotient depends only on the size of the residue field. The same holds for zeta functions counting dimensions of Hecke modules of intertwining operators associated to induced representations of such quotients.

20 pages, final version, to appear in the Journal of the LMS

Cited by in corpus (8)

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• The average size of the kernel of a matrix and orbits of linear groups
• Bivariate representation and conjugacy class zeta functions associated to unipotent group schemes, II: Groups of type F, G, and H
• The average size of the kernel of a matrix and orbits of linear groups, II: duality
• Bivariate representation and conjugacy class zeta functions associated to unipotent group schemes, I: Arithmetic properties
• On singularity properties of convolutions of algebraic morphisms
• Univariate and bivariate zeta functions of unipotent group schemes of type $G$
• On singularity properties of convolutions of algebraic morphisms -- the general case (with an appendix joint with Gady Kozma)