Rocky Mountain Journal of Mathematics

An identity for cocycles on coset spaces of locally compact groups

Abstract

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.

Article information

Source
Rocky Mountain J. Math., Volume 48, Number 1 (2018), 269-277.

Dates
First available in Project Euclid: 28 April 2018

https://projecteuclid.org/euclid.rmjm/1524880891

Digital Object Identifier
doi:10.1216/RMJ-2018-48-1-269

Mathematical Reviews number (MathSciNet)
MR3795743

Zentralblatt MATH identifier
06866710

Citation

Dharmadasa, H. Kumudini; Moran, William. An identity for cocycles on coset spaces of locally compact groups. Rocky Mountain J. Math. 48 (2018), no. 1, 269--277. doi:10.1216/RMJ-2018-48-1-269. https://projecteuclid.org/euclid.rmjm/1524880891

References

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