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Abstract

We prove here an identity for cocycles associated with homogeneous spaces in the context of locally compact groups. Mackey introduced cocycles (λ-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 λ is a real-valued Borel function on G/H × G satisfying the cocycle identity λ(x, st) = λ(x.s, t)λ(x, s), almost everywhere x ∈ G/H, s, t ∈ 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/Hx, G/Ky and G/(Hx ∩ Ky) for almost all x, y ∈ G. This also leads to an identity for modular functions of G and the corresponding subgroups.

Item Type:	Article
Authors/Creators:	Dharmadasa, HK and Moran, W
Keywords:	separable locally compact groups, modular function, quasi-invariant measure, lambda-function
Journal or Publication Title:	Rocky Mountain Journal of Mathematics
Publisher:	Rocky Mt Math Consortium
ISSN:	0035-7596
DOI / ID Number:	10.1216/RMJ-2018-48-1-269
Copyright Information:	Copyright 2018 Rocky Mountain Mathematics Consortium
Related URLs:	UTAS Profile: Dharmadasa, HK
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