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For any Hermitian Lie group $G$ of tube type we give a geometric quantizationprocedure 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 $K\subseteq G$ withcorresponding Cartan decomposition $\mathfrak{g}=\mathfrak{k}+\mathfrak{p}$ ofthe Lie algebra of $G$. We explicitly realize every such representation $\pi$on a Fock space consisting of square integrable holomorphic functions on itsassociated variety $Ass(\pi)\subseteq\mathfrak{p}_{\mathbb{C}}^*$.  The associated variety $Ass(\pi)$ is the closure of a single nilpotent$K_{\mathbb{C}}$-orbit$\mathcal{O}^{K_{\mathbb{C}}}\subseteq\mathfrak{p}_{\mathbb{C}}^*$ whichcorresponds by the Kostant-Sekiguchi correspondence to a nilpotent coadjoint$G$-orbit $\mathcal{O}^G\subseteq\mathfrak{g}^*$. The known Schr\"odinger modelof $\pi$ is a realization on $L^2(\mathcal{O})$, where$\mathcal{O}\subseteq\mathcal{O}^G$ is a Lagrangian submanifold. We constructan intertwining operator from the Schr\"odinger model to the new Fock model,the generalized Segal-Bargmann transform, which gives a geometric quantizationof 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 $K$-Besselfunctions on Jordan algebras which appear in the measure of$\mathcal{O}^{K_{\mathbb{C}}}$, as reproducing kernel of the Fock space and asintegral kernel of the Segal-Bargmann transform. As a corollary to ourconstruction we also obtain the integral kernel of the unitary inversionoperator in the Schr\"odinger model in terms of a multivariable $J$-Besselfunction as well as explicit Whittaker vectors.

A geometric quantization of the Kostant-Sekiguchi correpondence for scalar type unitary highest weight representations