§ 2. Homogeneous vector bundles and harmonic analysis 33 bundles. The representation is unitary if and only if the vector bundle F is unitary. We henceforth suppose that F is a unitary homogeneous vector bundle. If C (G; F 0 ) is the space of functions on G with values in F 0 , we consider its subspace C (G; ) = { f C (G; F 0 ) | f (ak) = (k) -1 f (a), for a G, k K } and we write ((a 1 )f )(a) = f (a -1 a), 1 for a, a 1 G and f C (G; ). Then is representation of G on the space C (G; ) and the mapping A : C (F ) C (G; ), defined by (Au)(a) = a -1 u(aK), for u C (F ) and a G, is an isomorphism of G-modules. In particular, if K is the subgroup {e} of G, where e is the identity element of G, and