Local integrability results in harmonic analysis on reductive groups in large positive characteristic

TL;DR

This paper extends Harish-Chandra's local integrability theorem for orbital integrals and characters from characteristic zero fields to large positive characteristic fields using motivic integration techniques.

Contribution

It proves that Fourier transforms of orbital integrals are specializations of motivic exponential functions, enabling transfer of integrability results to positive characteristic fields.

Findings

Orbital integrals' Fourier transforms are motivic exponential functions.

Harish-Chandra's theorem holds in large positive characteristic fields.

Local integrability of characters is established under certain conditions.

Abstract

Let $G$ be a connected reductive algebraic group over a non-Archimedean local field $K$, and let $g$ be its Lie algebra. By a theorem of Harish-Chandra, if $K$ has characteristic zero, the Fourier transforms of orbital integrals are represented on the set of regular elements in $g(K)$ by locally constant functions, which, extended by zero to all of $g(K)$, are locally integrable. In this paper, we prove that these functions are in fact specializations of constructible motivic exponential functions. Combining this with the Transfer Principle for integrability [R. Cluckers, J. Gordon, I. Halupczok, "Transfer principles for integrability and boundedness conditions for motivic exponential functions", preprint arXiv:1111.4405], we obtain that Harish-Chandra's theorem holds also when $K$ is a non-Archimedean local field of sufficiently large positive characteristic. Under the hypothesis on the existence of the mock exponential map, this also implies local integrability of Harish-Chandra characters of admissible representations of $G(K)$, where $K$ is an equicharacteristic field of sufficiently large (depending on the root datum of $G$) characteristic.