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2018 An identity for cocycles on coset spaces of locally compact groups
H. Kumudini Dharmadasa, William Moran
Rocky Mountain J. Math. 48(1): 269-277 (2018). DOI: 10.1216/RMJ-2018-48-1-269

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.

Citation

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H. Kumudini Dharmadasa. William Moran. "An identity for cocycles on coset spaces of locally compact groups." Rocky Mountain J. Math. 48 (1) 269 - 277, 2018. https://doi.org/10.1216/RMJ-2018-48-1-269

Information

Published: 2018
First available in Project Euclid: 28 April 2018

zbMATH: 06866710
MathSciNet: MR3795743
Digital Object Identifier: 10.1216/RMJ-2018-48-1-269

Subjects:
Primary: 22D30, 43A15, ‎43A65

Rights: Copyright © 2018 Rocky Mountain Mathematics Consortium

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Vol.48 • No. 1 • 2018
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