  For any Hermitian Lie group G of tube type we construct a Fock model of its
minimal representation. The Fock space is defined on the minimal nilpotent
K_C-orbit X in p_C and the L^2-inner product involves a K-Bessel function as
density. Here K is a maximal compact subgroup of G, and g_C=k_C+p_C is a
complexified Cartan decomposition. In this realization the space of k-finite
vectors consists of holomorphic polynomials on X. The reproducing kernel of the
Fock space is calculated explicitly in terms of an I-Bessel function. We
further find an explicit formula of a generalized Segal-Bargmann transform
which intertwines the Schroedinger and Fock model. Its kernel involves the same
I-Bessel function. Using the Segal--Bargmann transform we also determine the
integral kernel of the unitary inversion operator in the Schroedinger model
which is given by a J-Bessel function.
