Uniform cell decomposition with applications to Chevalley groups

TL;DR

This paper develops a uniform approach to integrals over definable sets in non-Archimedean fields and applies it to Chevalley groups, showing that certain zeta functions depend only on the residue field size for large primes.

Contribution

It extends Pas's results to express integrals uniformly and proves that zeta functions for Chevalley groups depend solely on the residue field size in large characteristic.

Findings

Zeta functions depend only on the residue field size for large primes.

Number of conjugacy classes in congruence quotients is determined by residue field size.

Dependence of zeta functions on residue field size is established for Hecke modules.

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.