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

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 $π$ on a Fock space consisting of square integrable holomorphic functions on its associated variety $Ass(π)\subseteq\mathfrak{p}_{\mathbb{C}}^*$. The associated variety $Ass(π)$ 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 Schrödinger model of $π$ 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 Schrödinger 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, Ørsted 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 Schrödinger model in terms of a multivariable $J$-Bessel function as well as explicit Whittaker vectors.

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