Motivic Haar measure on reductive groups

TL;DR

This paper introduces a motivic analogue of the Haar measure for reductive groups over formal Laurent series fields, using motivic integration to define an invariant measure in a non-locally compact setting.

Contribution

It constructs and proves the invariance of a motivic Haar measure for reductive groups over k((t)), extending Haar measure concepts beyond locally compact groups.

Findings

Defined a motivic Haar measure using motivic integration.

Proved the measure's invariance under group translations.

Showed the measure's independence from construction choices.

Abstract

We define a motivic analogue of the Haar measure for groups of the form G(k((t))), where k is an algebraically closed field of characteristic zero, and G is a reductive algebraic group defined over k. A classical Haar measure on such groups does not exist since they are not locally compact. We use the theory of motivic integration introduced by M. Kontsevich to define an additive function on a certain natural Boolean algebra of subsets of G(k((t))). This function takes values in the so-called dimensional completion of the Grothendieck ring of the category of varieties over the base field. It is invariant under translations by all elements of G(k((t))), and therefore we call it a motivic analogue of Haar measure. We give an explicit construction of the motivic Haar measure, and then prove that the result is independent of all the choices that are made in the process, even though we have no general uniqueness statement.