## Rocky Mountain Journal of Mathematics

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

### An identity for cocycles on coset spaces of locally compact groups

H. Kumudini Dharmadasa and William Moran

#### 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

**Permanent link to this document**

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

**Subjects**

Primary: 22D30: Induced representations 43A15: $L^p$-spaces and other function spaces on groups, semigroups, etc. 43A65: Representations of groups, semigroups, etc. [See also 22A10, 22A20, 22Dxx, 22E45]

**Keywords**

Separable locally compact group modular function quasi-invariant measure $\lambda $-function

#### 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