N strongly quasi invariant measure on double coset

TL;DR

This paper investigates the existence and properties of N-strongly quasi-invariant measures on double coset spaces in locally compact groups, establishing their connection to rho-functions and providing conditions for their existence.

Contribution

It introduces a framework linking N-strongly quasi-invariant measures with rho-functions on double coset spaces, expanding understanding of measure invariance in group theory.

Findings

Any N-strongly quasi-invariant measure arises from a rho-function

Conditions for measures to originate from rho-functions are characterized

Properties of rho-functions related to the structure of double coset spaces

Abstract

Let G be a locally compact group, H and K be two closed sub-groups of G, and N be the normalizer group of K in G. In this paper, the existence and properties of a rho-function for the triple (K,G,H) and an N-strongly quasi-invariant measure of double coset space K\G/H is investigated. In particular, it is shown that any such measure arises from a rho-function. Furthermore, the conditions under which an N-strongly quasi-invariant measure arises from a rho-function are studied.