An example is given of a
positive-definite measure μ on the group SL(2,R) which is extremal in the cone of
positive-definite measures, but the corresponding unitary representation Lμ is
reducible. By considering positive-definite distributions this anomaly disappears, and
for an arbitrary Lie group G and positive-definite distribution μ on G a
bijection is established between positive-definite distributions on G bounded by
μ and positive-definite intertwining operators for the representation Lμ.
As an application, cyclic vectors for Lμ are obtained by a simple explicit
construction.
