  For any Hermitian Lie group $G$ of tube type we give a geometric quantization
procedure of certain $K_{\mathbb{C}}$-orbits in $\mathfrak{p}_{\mathbb{C}}^*$
to obtain all scalar type highest weight representations. Here $K_{\mathbb{C}}$
is the complexification of a maximal compact subgroup $K\subseteq G$ with
corresponding Cartan decomposition $\mathfrak{g}=\mathfrak{k}+\mathfrak{p}$ of
the Lie algebra of $G$. We explicitly realize every such representation $\pi$
on a Fock space consisting of square integrable holomorphic functions on its
associated variety $Ass(\pi)\subseteq\mathfrak{p}_{\mathbb{C}}^*$.
  The associated variety $Ass(\pi)$ is the closure of a single nilpotent
$K_{\mathbb{C}}$-orbit
$\mathcal{O}^{K_{\mathbb{C}}}\subseteq\mathfrak{p}_{\mathbb{C}}^*$ which
corresponds by the Kostant-Sekiguchi correspondence to a nilpotent coadjoint
$G$-orbit $\mathcal{O}^G\subseteq\mathfrak{g}^*$. The known Schr\"odinger model
of $\pi$ is a realization on $L^2(\mathcal{O})$, where
$\mathcal{O}\subseteq\mathcal{O}^G$ is a Lagrangian submanifold. We construct
an intertwining operator from the Schr\"odinger model to the new Fock model,
the generalized Segal-Bargmann transform, which gives a geometric quantization
of the Kostant-Sekiguchi correspondence (a notion invented by Hilgert,
Kobayashi, {\O}rsted and the author).
  The main tool in our construction are multivariable $I$- and $K$-Bessel
functions on Jordan algebras which appear in the measure of
$\mathcal{O}^{K_{\mathbb{C}}}$, as reproducing kernel of the Fock space and as
integral kernel of the Segal-Bargmann transform. As a corollary to our
construction we also obtain the integral kernel of the unitary inversion
operator in the Schr\"odinger model in terms of a multivariable $J$-Bessel
function as well as explicit Whittaker vectors.
