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An identity for cocycles on coset spaces of locally compact groups

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Dharmadasa, HK ORCID: 0000-0003-2869-799X and Moran, W 2018 , 'An identity for cocycles on coset spaces of locally compact groups' , Rocky Mountain Journal of Mathematics, vol. 48, no. 1 , pp. 269-277 , doi: 10.1216/RMJ-2018-48-1-269.

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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

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