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For any Hermitian Lie group G of tube type we construct a Fock model of itsminimal representation. The Fock space is defined on the minimal nilpotentK_C-orbit X in p_C and the L^2-inner product involves a K-Bessel function asdensity. Here K is a maximal compact subgroup of G, and g_C=k_C+p_C is acomplexified Cartan decomposition. In this realization the space of k-finitevectors consists of holomorphic polynomials on X. The reproducing kernel of theFock space is calculated explicitly in terms of an I-Bessel function. Wefurther find an explicit formula of a generalized Segal-Bargmann transformwhich intertwines the Schroedinger and Fock model. Its kernel involves the sameI-Bessel function. Using the Segal--Bargmann transform we also determine theintegral kernel of the unitary inversion operator in the Schroedinger modelwhich is given by a J-Bessel function.

Fock model and Segal-Bargmann transform for minimal representations of Hermitian Lie groups