AN IDENTITY FOR COCYCLES ON COSET SPACES OF LOCALLY COMPACT GROUPS

作者：Dharmadasa H Kumudini^*; Moran William

来源：Rocky Mountain Journal of Mathematics, 2018, 48(1): 269-277.

DOI：10.1216/RMJ-2018-48-1-269

• Separable locally compact group
• modular function
• quasi-invariant measure
• lambda-function

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 x G satisfying the cocycle identity
lambda(x, st) = lambda(x.s, t)lambda(x, s),
almost everywhere x epsilon G/H, s, t epsilon 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 boolean AND K-y) for almost all x,y epsilon G. This also leads to an identity for modular functions of G and the corresponding subgroups.

• 出版日期2018