Pushforward measures on homogeneous spaces of non-unimodular groups and properties of modular functions

TL;DR

This paper derives a formula for pushforward measures on homogeneous spaces of non-unimodular groups using modular functions, revealing a key relation among these functions for certain subgroups.

Contribution

It provides a new explicit formula for pushforward measures on homogeneous spaces involving modular functions, extending understanding of measure theory in non-unimodular group contexts.

Findings

Derived a formula for pushforward measures using modular functions.

Established a key relation among modular functions of subgroups.

Extended measure theory to non-unimodular homogeneous spaces.

Abstract

This paper shows that a formula for the pushforward measures on the fiber bundle $ G \to G / H $ of homogeneous space of locally compact groups $ G $ and $ H $ can be written down by using the modular functions. As a result, we obtain the equality \begin{align} \frac{ \Delta_G ( h ) }{ \Delta_{ G / N } ( h N ) } = \frac{ \Delta_H ( h ) }{ \Delta_{ H / N } ( h N ) } \end{align} on the modular functions for any closed subgroups $ N $ and $ H $ of a locally compact group $ G $ with $ N \lhd G $ and $ N \subset H \subset G $ and any $ h \in H $.