N-Relatively Invariant and N-Invariant Measure on Double Coset Spaces

Abstract

Let G be a locally compact group, H and K be two closed subgroups 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 Nstrongly 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.