Integration on product spaces and GL_n of a valuation field over a local field

TL;DR

This paper develops a theory of translation-invariant integration on vector spaces and GL_n over valuation fields with local fields as residue fields, extending Fesenko's work to more general algebraic groups.

Contribution

It introduces a new framework for translation-invariant integration on algebraic groups over valuation fields, broadening the scope of previous theories.

Findings

Established integration theory on vector spaces over valuation fields.

Extended integration concepts to GL_n and arbitrary algebraic groups.

Built upon and generalized Fesenko's prior work.

Abstract

We present elements of a theory of translation-invariant integration on finite dimensional vector spaces and on GL_n over a valuation field with local field as residue field. We then discuss the case of an arbitrary algebraic group. This extends the work of Fesenko.