  We construct an explicit realization of a minimal representation of G, where
G is the conformal group of a real Jordan algebra N. We characterize spherical
vectors for these representation and prove that they are closely related to the
Bessel K-function $K_\tau (z)$. The resulting construction can be used to study
tensor powers of the minimal representation and establish an extension of the
Howe duality correspondence to some exceptional groups.
