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If a group G acts on a set S transitively on the right, then one can view S as G/K, where K is a subgroup of G. Thus a function on S can be considered as a function on G, invariant with respect to the right shifts by elements of K. It is especially interesting to consider bi-invariant functions on G since they can be identified with the functions on the set of G-orbits in S. If G is a locally compact group and K is its compact subgroup with Haar measures μG and μK respectively, then the set B ⊂ L1(G,μG) of all bi-invariant functions is an algebra with respect to the convolution and has a natural hypergroup structure related to generalized translation operators (see the survey [23] and the references given there):

Quantum Stiefel manifold and double cosets of quantum unitary group