# Measure and Integration on $GL_2$ over a Two-Dimensional Local Field

## TL;DR

This paper develops a translation-invariant measure and integral on the group $GL_2$ over a two-dimensional local field, extending classical and Fesenko's theories, and reconciling with Morrow's prior integral definition.

## Contribution

It introduces a new measure and integral on $GL_2$ over a two-dimensional local field, combining classical and Fesenko's theories, and aligns with Morrow's earlier work.

## Key findings

- Defined a translation-invariant measure and integral on $GL_2$ over a two-dimensional local field.
- Provided multiple expressions for the integral, including one matching Morrow's previous definition.
- Extended classical $GL_2$ theory to the setting of two-dimensional local fields.

## Abstract

We define a translation-invariant measure and integral on $GL_2$ over a two-dimensional local field $F$ by combining elements of the classical $GL_2$ theory and the theory developed by Fesenko for the field $F$ itself. We give several alternate expressions for the integral, including one which agrees with the integral defined previously by Morrow.