The canonical measure on a reductive p-adic group is motivic

TL;DR

This paper proves that the canonical Haar measure on a reductive p-adic group is motivic, using definability of parahoric subgroups in the Denef-Pas language, which simplifies previous proofs and links measures to motivic functions.

Contribution

It establishes the motivicity of the canonical Haar measure on reductive p-adic groups by proving definability of parahoric subgroups and related structures in the Denef-Pas language.

Findings

Parahoric subgroups are definable in the Denef-Pas language.

The canonical Haar measure is motivic.

Formal degrees of certain representations are motivic functions.

Abstract

Let $G$ be a connected reductive group over a non-Archimedean local field. We prove that its parahoric subgroups are definable in the Denef-Pas language, which is a first-order language of logic used in the theory of motivic integration developed by Cluckers and Loeser. The main technical result is the definability of the connected component of the N\'eron model of a tamely ramified algebraic torus. As a corollary, we prove that the canonical Haar measure on $G$, which assigns volume $1$ to the particular \emph{canonical} maximal parahoric defined by Gross, is motivic. This result resolves a technical difficulty that arose in Cluckers-Gordon-Halupczok and Shin-Templier and permits a simplification of some of the proofs in those articles. It also allows us to show that formal degree of a compactly induced representation is a motivic function of the parameters defining the representation.