Project Euclid

We prove here an identity for cocycles associated with homogeneous spaces in the context of locally compact groups. Mackey introduced cocycles ($\lambda $-functions) in his work on representation theory of such groups. For a given locally compact group $G$ and a closed subgroup $H$ of $G$, with right coset space $G/H$, a cocycle $\lambda $ is a real-valued Borel function on $G/H \times G$ satisfying the cocycle identity \[ \lambda (x, st)=\lambda (x.s,t)\lambda (x,s), \] \[\mbox {almost everywhere } x\in G/H,\ s,t\in G, \] where the ``almost everywhere" is with respect to a measure whose null sets pull back to Haar measure null sets on $G$. Let $H$ and $K$ be regularly related closed subgroups of $G.$ Our identity describes a relationship among cocycles for $G/H^x$, $G/K^y$ and $G/(H^x\cap K^y)$ for almost all $x,y\in G$. This also leads to an identity for modular functions of $G$ and the corresponding subgroups.