N strongly quasi invariant measure on double coset

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.